qu.1.topic=Basic Matrix Operations@

qu.1.1.mode=Inline@
qu.1.1.name=Linear Algebra - Identities - T@
qu.1.1.comment=@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=@
qu.1.1.uid=4f4e5449-bd8d-40e4-a388-05e017af9a06@
qu.1.1.info=  Course=Introductory Mathematical Economics;
  Topic=Linear Algebra Identities;
  Sub-Topic=Transpose;
  Author=Asha Sadanand;
  Feature=Multiple Choice;
@
qu.1.1.weighting=1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.name=sro_id_1@
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.fixed=@
qu.1.1.part.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>B</mi><mrow><mi>T</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>A</mi></mrow></mstyle></math>@
qu.1.1.part.1.question=null@
qu.1.1.part.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>B</mi><mrow><mi>T</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>A</mi><mrow><mi>T</mi></mrow></msup></mrow></mstyle></math>@
qu.1.1.part.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>A</mi><mrow><mi>T</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>B</mi><mrow><mi>T</mi></mrow></msup></mrow></mstyle></math>@
qu.1.1.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>B</mi><mrow><mi>T</mi></mrow></msup></mrow></mstyle></math>@
qu.1.1.part.1.mode=Multiple Choice@
qu.1.1.part.1.display=vertical@
qu.1.1.part.1.answer=3@
qu.1.1.question=<p>Suppose&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> are two non-singular matrices.&nbsp;&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>B</mi></mrow></mfenced><mrow><mi>T</mi></mrow></msup></mrow></mstyle></math><span>&nbsp;</span> is equal to<1><span> </span></p>@

qu.1.2.mode=Inline@
qu.1.2.name=Linear Algebra - Matrix equivalences - Inv and T@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=@
qu.1.2.uid=77192abe-3ade-471f-b97f-a4b7e426bf99@
qu.1.2.info=  Course=Introductory Mathematical Economics;
  Topic=Linear Algebra Identities;
  Sub-Topic=Inverse And Transpose;
  Author=Asha Sadanand;
  Feature=Multiple Choice;
@
qu.1.2.weighting=1@
qu.1.2.numbering=alpha@
qu.1.2.part.1.name=sro_id_1@
qu.1.2.part.1.editing=useHTML@
qu.1.2.part.1.fixed=@
qu.1.2.part.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>B</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>A</mi><mrow><mi>T</mi></mrow></msup></mrow></mstyle></math>@
qu.1.2.part.1.question=null@
qu.1.2.part.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>B</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>A</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mfenced><mrow><mi>T</mi></mrow></msup></mrow></mstyle></math>@
qu.1.2.part.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>A</mi><mrow><mi>T</mi></mrow></msup></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>B</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>@
qu.1.2.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>A</mi><mrow><mi>T</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>B</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>@
qu.1.2.part.1.mode=Multiple Choice@
qu.1.2.part.1.display=vertical@
qu.1.2.part.1.answer=3@
qu.1.2.question=<p>Suppose&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> are two non-singular matrices.&nbsp;&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>A</mi><mrow><mi>T</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>B</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math><span>&nbsp;</span> is equal to<1><span> </span></p>@

qu.1.3.mode=Inline@
qu.1.3.name=Matrix Multiplication - Simple@
qu.1.3.comment=<p>AB=$anspretty</p>@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$a=range(2,3);
$b=range(2,3);
$e=range(2,3);
$c=$a*$e;
$d=1/$c+0.000001;
$v=maple("
randomize():
with(LinearAlgebra):
v1:=RandomMatrix($a,$b,generator=(-3..3)):
v2:=RandomMatrix($b,$e,generator=(-3..3)):
v3:=v1.v2:
v4:=MathML[ExportPresentation](v1):
v5:=MathML[ExportPresentation](v2):
v6:=MathML[ExportPresentation](v3):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(v5,string),convert(v6,string)
");
$A=switch(3,$v);
$B=switch(4,$v);
$ans=switch(2,$v);
$anspretty=switch(5,$v);@
qu.1.3.uid=f737fd41-1741-4d39-b660-7603edc20f2b@
qu.1.3.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Matrix Multiplication;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Students Use Equation Editor;
@
qu.1.3.weighting=1@
qu.1.3.numbering=alpha@
qu.1.3.part.1.name=sro_id_1@
qu.1.3.part.1.maple_answer=printf("$anspretty");@
qu.1.3.part.1.editing=useHTML@
qu.1.3.part.1.question=(Unset)@
qu.1.3.part.1.libname=@
qu.1.3.part.1.mode=Maple@
qu.1.3.part.1.allow2d=2@
qu.1.3.part.1.plot=@
qu.1.3.part.1.maple=ans:=$ans:
grade:=0:
for i from 1 to $a do
for j from 1 to $e do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+$d:
end if;
end;
end;
grade;@
qu.1.3.part.1.type=maple@
qu.1.3.question=<p>Compute A times B.</p><p>A=$A&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; B=$B</p><p>&nbsp;</p><p>&nbsp;</p><p>(To input your answer, right-click on the box below to  bring up the symbols bar, select the button with a square made out of 9  smaller squares and select the appropriate size of matrix. If the correct size is not shown, select&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mi>m</mi></mrow></mstyle></math>and set the dimensions yourself. )</p><p>AB=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.4.mode=Inline@
qu.1.4.name=Transpose of a Matrix@
qu.1.4.comment=@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$a=range(2,4);
$b=range(2,4);
$c=$a*$b;
$d=1/$c+0.000001;
$v=maple("
randomize():
v1:=LinearAlgebra[RandomMatrix]($a,$b,generator=rand(-9..9)):
v2:=MathML[ExportPresentation](v1):
v3:=LinearAlgebra[Transpose](v1):
v4:=MathML[ExportPresentation](v3):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string)
");
$A=switch(0,$v);
$Apretty=switch(1,$v);
$AT=switch(2,$v);
$ATpretty=switch(3,$v);@
qu.1.4.uid=fd1b4b82-07f7-4be2-9db3-b4f21e08f105@
qu.1.4.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Transpose Of A Matrix;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Students Use Equation Editor;
@
qu.1.4.weighting=1@
qu.1.4.numbering=alpha@
qu.1.4.part.1.name=sro_id_1@
qu.1.4.part.1.maple_answer=printf("$ATpretty");@
qu.1.4.part.1.editing=useHTML@
qu.1.4.part.1.question=(Unset)@
qu.1.4.part.1.libname=@
qu.1.4.part.1.mode=Maple@
qu.1.4.part.1.allow2d=2@
qu.1.4.part.1.plot=@
qu.1.4.part.1.maple=ans:=$AT:
grade:=0:
for i from 1 to $b do
for j from 1 to $a do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+$d:
end if;
end;
end;
grade;@
qu.1.4.part.1.type=maple@
qu.1.4.question=<p>Find the transpose of the following matrix:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$Apretty</p><p>&nbsp;</p><p>&nbsp;</p><p>(To input your answer, right-click on the box below to  bring up the  symbols bar, select the button with a square made out of 9  smaller  squares and select the appropriate size of matrix. If the correct size  is not shown, select&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mi>m</mi></mrow></mstyle></math> and set the dimensions yourself. )</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>A</mi><mrow><mi>T</mi></mrow></msup></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.5.mode=Inline@
qu.1.5.name=Linear Algebra - Matrix equivalences - det and T@
qu.1.5.comment=@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=@
qu.1.5.uid=b7ba8270-9ae3-408a-afa5-4bc6dd7f6420@
qu.1.5.info=  Course=Introductory Mathematical Economics;
  Topic=Linear Algebra Identities;
  Sub-Topic=Determinant And Transpose;
  Author=Asha Sadanand;
  Feature=Multiple Choice;
@
qu.1.5.weighting=1@
qu.1.5.numbering=alpha@
qu.1.5.part.1.name=sro_id_1@
qu.1.5.part.1.editing=useHTML@
qu.1.5.part.1.fixed=@
qu.1.5.part.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>det</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>det</mi><msup><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><mrow><mi>T</mi></mrow></msup></mrow></mstyle></math>@
qu.1.5.part.1.question=null@
qu.1.5.part.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>det</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>det</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>A</mi><mrow><mi>T</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>@
qu.1.5.part.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>det</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>det</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced></mrow></mstyle></math>@
qu.1.5.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>det</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>B</mi><mrow><mi>T</mi></mrow></msup></mrow><mrow><mi></mi></mrow></mstyle></math>@
qu.1.5.part.1.mode=Multiple Choice@
qu.1.5.part.1.display=vertical@
qu.1.5.part.1.answer=2@
qu.1.5.question=<p>Suppose&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> are two square matrices. Then, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>det</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>B</mi><mrow><mi>T</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math><span>&nbsp;</span> is equal to<1><span> </span></p>@

qu.1.6.mode=Inline@
qu.1.6.name=Matrix Multiplication - Importance of Order@
qu.1.6.comment=<p>AB=$ABpretty</p>
<p>BA=$BApretty</p>@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=$a=range(2,3);
$v=maple("
randomize():
with(LinearAlgebra):
if $a=2 then b:=3
else b:=2
end if:
v1:=RandomMatrix($a,b,generator=(-3..3)):
v2:=RandomMatrix(b,$a,generator=(-3..3)):
v3:=v1.v2:
v4:=MathML[ExportPresentation](v1):
v5:=MathML[ExportPresentation](v2):
v6:=MathML[ExportPresentation](v3):
v7:=v2.v1:
v8:=MathML[ExportPresentation](v7):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(v5,string),convert(v6,string),convert(v7,string),convert(v8,string)
");
$A=switch(3,$v);
$B=switch(4,$v);
$AB=switch(2,$v);
$ABpretty=switch(5,$v);
$BA=switch(6,$v);
$BApretty=switch(7,$v);@
qu.1.6.uid=546116f2-fded-4cb9-9403-93c083c48da2@
qu.1.6.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Matrix Multiplication - Order;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Students Use Equation Editor;
@
qu.1.6.weighting=1,1@
qu.1.6.numbering=alpha@
qu.1.6.part.1.name=sro_id_1@
qu.1.6.part.1.maple_answer=printf("$ABpretty");@
qu.1.6.part.1.editing=useHTML@
qu.1.6.part.1.question=(Unset)@
qu.1.6.part.1.libname=@
qu.1.6.part.1.mode=Maple@
qu.1.6.part.1.allow2d=2@
qu.1.6.part.1.plot=@
qu.1.6.part.1.maple=ans:=$AB:
grade:=0:
if $a=2 then
for i from 1 to 2 do
for j from 1 to 2 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.25:
end if;
end;
end;
else 
for i from 1 to 3 do
for j from 1 to 3 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.111112:
end if;
end;
end;
end if;
grade;@
qu.1.6.part.1.type=maple@
qu.1.6.part.2.name=sro_id_2@
qu.1.6.part.2.maple_answer=printf("$BApretty");@
qu.1.6.part.2.editing=useHTML@
qu.1.6.part.2.question=(Unset)@
qu.1.6.part.2.libname=@
qu.1.6.part.2.mode=Maple@
qu.1.6.part.2.allow2d=2@
qu.1.6.part.2.plot=@
qu.1.6.part.2.maple=ans:=$BA:
grade:=0:
if $a=3 then
for i from 1 to 2 do
for j from 1 to 2 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.25:
end if;
end;
end;
else 
for i from 1 to 3 do
for j from 1 to 3 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.111112:
end if;
end;
end;
end if;
grade;@
qu.1.6.part.2.type=maple@
qu.1.6.question=<p>A=$A&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; B=$B</p><p>&nbsp;</p><p>Compute A times B.</p><p>&nbsp;</p><p>(To input your answer, right-click on the box below to  bring up the symbols bar, select the button with a square made out of 9  smaller squares and select the appropriate size of matrix. If the correct size is not shown, select&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mi>m</mi></mrow></mstyle></math>and set the dimensions yourself. )</p><p>AB=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>Now find B times A.</span></p><p>&nbsp;</p><p><span>BA=</span><span>&nbsp;</span><2><span>&nbsp;</span></p>@

qu.1.7.mode=Inline@
qu.1.7.name=Trace of a Matrix@
qu.1.7.comment=<p>The trace is the sum of the diagonals.&nbsp;</p>
<p>&nbsp;</p>
<p>Therefore, the trace of $Apretty is $trace.</p>@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$i=range(3,5);
$m=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix]($i,generator=rand(-20..20),attributes=[nonsingular]):
m2:=LinearAlgebra[Trace](m1):
m3:=MathML[ExportPresentation](m1):
m2,convert(m3,string)
");
$trace=switch(0,$m);
$Apretty=switch(1,$m);@
qu.1.7.uid=754d62a0-afb3-4b26-94f9-3f4165c54f2b@
qu.1.7.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Trace;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Students Use Equation Editor;
@
qu.1.7.weighting=1@
qu.1.7.numbering=alpha@
qu.1.7.part.1.name=sro_id_1@
qu.1.7.part.1.answer.units=@
qu.1.7.part.1.numStyle=   @
qu.1.7.part.1.editing=useHTML@
qu.1.7.part.1.showUnits=false@
qu.1.7.part.1.question=(Unset)@
qu.1.7.part.1.mode=Numeric@
qu.1.7.part.1.grading=exact_value@
qu.1.7.part.1.negStyle=both@
qu.1.7.part.1.answer.num=$trace@
qu.1.7.question=<p>What is the trace of the following matrix?</p><p>&nbsp;</p><p>A=$Apretty</p><p>&nbsp;</p><p>trace(A)=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p>@

qu.1.8.mode=Inline@
qu.1.8.name=Linear Algebra - Matrix equivalences - Inv of a sum@
qu.1.8.comment=@
qu.1.8.editing=useHTML@
qu.1.8.solution=@
qu.1.8.algorithm=@
qu.1.8.uid=592e0783-102d-46a7-9818-ce5b6e246eec@
qu.1.8.info=  Course=Introductory Mathematical Economics;
  Topic=Linear Algebra Identities;
  Sub-Topic=Inverse And Addition;
  Author=Asha Sadanand;
  Feature=Multiple Choice;
@
qu.1.8.weighting=1@
qu.1.8.numbering=alpha@
qu.1.8.part.1.name=sro_id_1@
qu.1.8.part.1.editing=useHTML@
qu.1.8.part.1.fixed=@
qu.1.8.part.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mfrac><mn>1</mn><mrow><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>B</mi></mrow></mfenced></mrow></mfrac></mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>@
qu.1.8.part.1.question=null@
qu.1.8.part.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>B</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>It</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi>cannot</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>be</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>simplified</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math>@
qu.1.8.part.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>A</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>B</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>2</mn><msup><mfenced open='(' close=')' separators=','><mrow><mi>AB</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>@
qu.1.8.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>B</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>@
qu.1.8.part.1.mode=Multiple Choice@
qu.1.8.part.1.display=vertical@
qu.1.8.part.1.answer=3@
qu.1.8.question=<p>&nbsp;</p><p>Suppose&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> are two non-singular matrices, and their sum is also non-singular.&nbsp;&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>B</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math><span>&nbsp;</span> is equal to<1><span> </span></p>@

qu.1.9.mode=Inline@
qu.1.9.name=Matrix Multiplication - No Numbers@
qu.1.9.comment=<p>AB=$anspretty</p>@
qu.1.9.editing=useHTML@
qu.1.9.solution=@
qu.1.9.algorithm=$v=maple("
randomize():

v1:=Matrix([[a,b],[c,d]]):
v2:=Matrix([[e,f],[g,h]]):
v3:=v1.v2:
v4:=MathML[ExportPresentation](v1):
v5:=MathML[ExportPresentation](v2):
v6:=MathML[ExportPresentation](v3):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(v5,string),convert(v6,string)
");
$A=switch(3,$v);
$B=switch(4,$v);
$ans=switch(2,$v);
$anspretty=switch(5,$v);@
qu.1.9.uid=ecc15d7c-3db6-4c25-97e2-61038fd3a588@
qu.1.9.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Matrix Multiplication - No Numbers;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Students Use Equation Editor;
@
qu.1.9.weighting=1@
qu.1.9.numbering=alpha@
qu.1.9.part.1.name=sro_id_1@
qu.1.9.part.1.maple_answer=printf("$anspretty");@
qu.1.9.part.1.editing=useHTML@
qu.1.9.part.1.question=(Unset)@
qu.1.9.part.1.libname=@
qu.1.9.part.1.mode=Maple@
qu.1.9.part.1.allow2d=2@
qu.1.9.part.1.plot=@
qu.1.9.part.1.maple=ans:=$ans:
LinearAlgebra[Equal](ans,$RESPONSE)@
qu.1.9.part.1.type=maple@
qu.1.9.question=<p>Find an expression for A times B.</p><p>A=$A&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; B=$B</p><p>&nbsp;</p><p>&nbsp;</p><p>(To input your answer, right-click on the box below to  bring up the symbols bar, select the button with a square made out of 9  smaller squares and select the appropriate size of matrix. If the correct size is not shown, select&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mi>m</mi></mrow></mstyle></math>and set the dimensions yourself. )</p><p>AB=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.10.mode=Inline@
qu.1.10.name=Matrix Addition/Subtraction@
qu.1.10.comment=<p>$A $operator $B = $anspretty</p>@
qu.1.10.editing=useHTML@
qu.1.10.solution=@
qu.1.10.algorithm=$a=range(2,4);
$b=range(2,4);
$c=$a*$b;
$d=1/$c+.001;
$k=range(0,1);
$v=maple("
randomize():
with(LinearAlgebra):
v1:=RandomMatrix($a,$b,generator=(-5..5)):
v2:=RandomMatrix($a,$b,generator=(-5..5)):
v3:=v1+v2:
v4:=MathML[ExportPresentation](v1):
v5:=MathML[ExportPresentation](v2):
v6:=MathML[ExportPresentation](v3):
v7:=v1-v2:
v8:=MathML[ExportPresentation](v7):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(v5,string),convert(v6,string),convert(v7,string),convert(v8,string)
");
$add=switch(2,$v);
$addpretty=switch(5,$v);
$subtract=switch(6,$v);
$subtractpretty=switch(7,$v);
$A=switch(3,$v);
$B=switch(4,$v);
$ans=switch($k,"$add","$subtract");
$operator=switch($k,'+','-');
$anspretty=switch($k,"$addpretty","$subtractpretty");@
qu.1.10.uid=61d11e6e-3600-42fd-a34b-53763914a21a@
qu.1.10.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Addition/Subtraction;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Students Use Equation Editor;
@
qu.1.10.weighting=1@
qu.1.10.numbering=alpha@
qu.1.10.part.1.name=sro_id_1@
qu.1.10.part.1.maple_answer=printf("$anspretty");@
qu.1.10.part.1.editing=useHTML@
qu.1.10.part.1.question=(Unset)@
qu.1.10.part.1.libname=@
qu.1.10.part.1.mode=Maple@
qu.1.10.part.1.allow2d=2@
qu.1.10.part.1.plot=@
qu.1.10.part.1.maple=ans:=$ans:
grade:=0:
for i from 1 to $a do
for j from 1 to $b do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+$d:
end if;
end;
end;
grade;@
qu.1.10.part.1.type=maple@
qu.1.10.question=<p>Calculate the following:</p><p>&nbsp;</p><p>(To input your answer, right-click on the box below to  bring up the symbols bar, select the button with a square made out of 9  smaller squares and select the appropriate size of matrix. If the correct size is not shown, select <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mi>m</mi></mrow></mstyle></math>and set the dimensions yourself. )&nbsp;</p><p>$A $operator $B=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.11.mode=Inline@
qu.1.11.name=Linear Algebra - Identities - Inv@
qu.1.11.comment=@
qu.1.11.editing=useHTML@
qu.1.11.solution=@
qu.1.11.algorithm=@
qu.1.11.uid=0fcc0371-55f5-47e0-957b-4cfee090232c@
qu.1.11.info=  Course=Introductory Mathematical Economics;
  Topic=Linear Algebra Identities;
  Sub-Topic=Inverse;
  Author=Asha Sadanand;
  Feature=Multiple Choice;
@
qu.1.11.weighting=1@
qu.1.11.numbering=alpha@
qu.1.11.part.1.name=sro_id_1@
qu.1.11.part.1.editing=useHTML@
qu.1.11.part.1.fixed=@
qu.1.11.part.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>B</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>A</mi></mrow></mstyle></math>@
qu.1.11.part.1.question=null@
qu.1.11.part.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>B</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>A</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>@
qu.1.11.part.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>A</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>B</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>@
qu.1.11.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>B</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>@
qu.1.11.part.1.mode=Multiple Choice@
qu.1.11.part.1.display=vertical@
qu.1.11.part.1.answer=3@
qu.1.11.question=<p>Suppose&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> are two non-singular matrices.&nbsp;&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>B</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math><span>&nbsp;</span> is equal to<1><span> </span></p>@

qu.1.12.mode=Inline@
qu.1.12.name=Matrix Multiplication - might not conform@
qu.1.12.comment=<p>AB=$anspretty</p>@
qu.1.12.editing=useHTML@
qu.1.12.hint.1=The inner dimensions need to match for the matrices to conform.@
qu.1.12.solution=@
qu.1.12.algorithm=$a=range(2,3);
$b=range(2,3);
$d=range(2,3);
$e=range(1,3);
$c=1/($a*$e);
$v=maple("
randomize():
with(LinearAlgebra):
v1:=RandomMatrix($a,$b,generator=(-3..3)):
v2:=RandomMatrix($d,$e,generator=(-3..3)):
if $b = $d then v3:=v1.v2
else v3:='DNC'
end if:
v4:=MathML[ExportPresentation](v1):
v5:=MathML[ExportPresentation](v2):
v6:=MathML[ExportPresentation](v3):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(v5,string),convert(v6,string)
");
$A=switch(3,$v);
$B=switch(4,$v);
$ans=switch(2,$v);
$anspretty=switch(5,$v);@
qu.1.12.uid=24c37a56-dc38-4419-ba9c-9c3112a02e5d@
qu.1.12.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Matrix Multiplication;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Students Use Equation Editor;
@
qu.1.12.weighting=1@
qu.1.12.numbering=alpha@
qu.1.12.part.1.name=sro_id_1@
qu.1.12.part.1.maple_answer=printf("$anspretty");@
qu.1.12.part.1.editing=useHTML@
qu.1.12.part.1.question=(Unset)@
qu.1.12.part.1.libname=@
qu.1.12.part.1.mode=Maple@
qu.1.12.part.1.allow2d=2@
qu.1.12.part.1.plot=@
qu.1.12.part.1.maple=ans:=$ans:
gradePer:=1/(LinearAlgebra[RowDimension](ans)*LinearAlgebra[ColumnDimension](ans));
if (ans='DNC') then 
StringTools[CompareCI](ans,$RESPONSE);
else 
grade:=0:
$RESPONSE:=convert($RESPONSE,Matrix):
for i from 1 to $a do
for j from 1 to $e do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+gradePer:
end if;
end;
end;
grade;
end if;
@
qu.1.12.part.1.type=maple@
qu.1.12.question=<p>Find AB.</p><p>A=$A&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; B=$B</p><p>If the matrices do not conform, type DNC.</p><p>&nbsp;(To input your answer, right-click on the box below to bring up the symbols bar, select the button with a square made out of 9 smaller squares and select the appropriate size of matrix. If the correct size is not shown, select <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mi>m</mi></mrow></mstyle></math>and set the dimensions yourself. )</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p><span>Note:&nbsp; Please enter the values exactly - ie. as fractions or whole numbers, not&nbsp; as decimal numbers.</span></p>@

qu.1.13.mode=Inline@
qu.1.13.name=Matrix Addition/Subtraction - with scalar multiplication@
qu.1.13.comment=<p>$c$A $operator $d$B = $anspretty</p>@
qu.1.13.editing=useHTML@
qu.1.13.solution=@
qu.1.13.algorithm=$a=range(2,4);
$b=range(2,4);
$c=range(2,9);
$d=range(2,9);
$k=range(0,1);
$v=maple("
randomize():
with(LinearAlgebra):
v1:=RandomMatrix($a,$b,generator=(-5..5)):
v2:=RandomMatrix($a,$b,generator=(-5..5)):
v3:=$c*v1+$d*v2:
v4:=MathML[ExportPresentation](v1):
v5:=MathML[ExportPresentation](v2):
v6:=MathML[ExportPresentation](v3):
v7:=$c*v1-$d*v2:
v8:=MathML[ExportPresentation](v7):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(v5,string),convert(v6,string),convert(v7,string),convert(v8,string)
");
$add=switch(2,$v);
$addpretty=switch(5,$v);
$subtract=switch(6,$v);
$subtractpretty=switch(7,$v);
$A=switch(3,$v);
$B=switch(4,$v);
$ans=switch($k,"$add","$subtract");
$operator=switch($k,'+','-');
$anspretty=switch($k,"$addpretty","$subtractpretty");@
qu.1.13.uid=11a1317d-f14f-4f52-8051-0e027896f597@
qu.1.13.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Addition/Subtraction;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Students Use Equation Editor;
@
qu.1.13.weighting=1@
qu.1.13.numbering=alpha@
qu.1.13.part.1.name=sro_id_1@
qu.1.13.part.1.maple_answer=printf("$anspretty");@
qu.1.13.part.1.editing=useHTML@
qu.1.13.part.1.question=(Unset)@
qu.1.13.part.1.libname=@
qu.1.13.part.1.mode=Maple@
qu.1.13.part.1.allow2d=2@
qu.1.13.part.1.plot=@
qu.1.13.part.1.maple=ans:=$ans:
LinearAlgebra[Equal](ans,$RESPONSE)@
qu.1.13.part.1.type=maple@
qu.1.13.question=<p>Calculate the following:</p><p>&nbsp;</p><p>&nbsp;(To input your answer, right-click on the box below to  bring up the symbols bar, select the button with a square made out of 9  smaller squares and select the appropriate size of matrix. If the correct size is not shown, select <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mi>m</mi></mrow></mstyle></math>and set the dimensions yourself. )</p><p>$c$A $operator $d$B=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.2.topic=Inverses@

qu.2.1.mode=Inline@
qu.2.1.name=Det of a 3x3 matrix, Invertibility@
qu.2.1.comment=<p>The determinant of $A is $det.</p>
<p>Because the determinant $feedback2, A $feedback invertible.</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$m=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](3,generator=rand(-1..2)):
m2:=LinearAlgebra[Determinant](m1):
m4:=MathML[ExportPresentation](m1):
if m2=0 then m3:=0:
else m3:=1:
end if:
m2,convert(m4,string),m3
");
$det=switch(0,$m);
$A=switch(1,$m);
$k=switch(2,$m);
$ANSWER=switch($k,'No','Yes');
$WRONG=switch($k,'Yes','No');
$feedback=switch($k,'is not','is');
$feedback2=switch($k,'is equal to zero','is not equal to zero');@
qu.2.1.uid=82178e97-71e5-4f2a-9bcc-61f8307409d0@
qu.2.1.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants, Invertibility;
  Author=Katherine Dare;
  Difficulty=Easy;
@
qu.2.1.weighting=1,1@
qu.2.1.numbering=alpha@
qu.2.1.part.1.name=sro_id_1@
qu.2.1.part.1.answer.units=@
qu.2.1.part.1.numStyle=   @
qu.2.1.part.1.editing=useHTML@
qu.2.1.part.1.showUnits=false@
qu.2.1.part.1.question=(Unset)@
qu.2.1.part.1.mode=Numeric@
qu.2.1.part.1.grading=exact_value@
qu.2.1.part.1.negStyle=both@
qu.2.1.part.1.answer.num=$det@
qu.2.1.part.2.grader=exact@
qu.2.1.part.2.name=sro_id_2@
qu.2.1.part.2.editing=useHTML@
qu.2.1.part.2.display.permute=true@
qu.2.1.part.2.answer.3=Not Enough Info@
qu.2.1.part.2.question=(Unset)@
qu.2.1.part.2.answer.2=$WRONG@
qu.2.1.part.2.answer.1=$ANSWER@
qu.2.1.part.2.mode=List@
qu.2.1.part.2.display=menu@
qu.2.1.part.2.credit.3=0.0@
qu.2.1.part.2.credit.2=0.0@
qu.2.1.part.2.credit.1=1.0@
qu.2.1.question=<p>What is the determinant of the following matrix?</p><p>A=$A</p><p>det(A)=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>Is A invertible?</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p>@

qu.2.2.mode=Inline@
qu.2.2.name=Inv of a 2x2 matrix without numbers@
qu.2.2.comment=<p>The determinant is $det.</p>
<p>The inverse is $invpretty.</p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$m=maple("
m1:=Matrix([[a, b], [c, d]]):
m2:=Matrix(2, 2, [['d'/('a'*'d'-'b'*'c'),-'b'/('a'*'d'-'b'*'c')],[-'c'/('a'*'d'-'b'*'c'),'a'/('a'*'d'-'b'*'c')]]):
m3:=MathML[ExportPresentation](m2):
m4:=MathML[ExportPresentation](m1):
m5:=LinearAlgebra[Determinant](m1):
convert(m2,string),convert(m3,string),convert(m4,string),convert(m5,string)
");
$inv=switch(0,$m);
$invpretty=switch(1,$m);
$A=switch(2,$m);
$det=switch(3,$m);@
qu.2.2.uid=8d29ee47-610c-41c3-8ceb-9f10e2bbaf7f@
qu.2.2.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Inverses;
  Author=Katherine Dare;
  Difficulty=Hard;
  Feature=Students Use Equation Editor;
@
qu.2.2.weighting=1@
qu.2.2.numbering=alpha@
qu.2.2.part.1.name=sro_id_1@
qu.2.2.part.1.maple_answer=printf("$invpretty");@
qu.2.2.part.1.editing=useHTML@
qu.2.2.part.1.question=(Unset)@
qu.2.2.part.1.libname=@
qu.2.2.part.1.mode=Maple@
qu.2.2.part.1.allow2d=2@
qu.2.2.part.1.plot=@
qu.2.2.part.1.maple=ans:=$inv:
grade:=0:
for i from 1 to 2 do
for j from 1 to 2 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.25:
end if;
end;
end;
grade;@
qu.2.2.part.1.type=maple@
qu.2.2.question=<p>What is the inverse of A?</p><p>&nbsp;A=$A</p><p>&nbsp;</p><p>(To input your answer, right-click on the box below to  bring up the symbols bar, select the button with a square made out of 9  smaller squares and select the appropriate size of matrix. If the correct size is not shown, select <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mi>m</mi></mrow></mstyle></math>and set the dimensions yourself. )</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p>@

qu.2.3.mode=Inline@
qu.2.3.name=Simple A*Ainverse question@
qu.2.3.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>A</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>= $Ipretty if A is $b<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow></mstyle></math>$b.</p>@
qu.2.3.editing=useHTML@
qu.2.3.solution=@
qu.2.3.algorithm=$b=range(2,4);
$m=maple("
m1:=Matrix($b, $b, shape = identity):
m2:=MathML[ExportPresentation](m1):
convert(m1,string),convert(m2,string)
");
$I=switch(0,$m);
$Ipretty=switch(1,$m);@
qu.2.3.uid=173c5bdc-03eb-47ad-8ff2-056570f7c5d6@
qu.2.3.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Identity Matrix;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Students Use Equation Editor;
@
qu.2.3.weighting=1@
qu.2.3.numbering=alpha@
qu.2.3.part.1.name=sro_id_1@
qu.2.3.part.1.maple_answer=printf("$Ipretty");@
qu.2.3.part.1.editing=useHTML@
qu.2.3.part.1.question=(Unset)@
qu.2.3.part.1.libname=@
qu.2.3.part.1.mode=Maple@
qu.2.3.part.1.allow2d=2@
qu.2.3.part.1.plot=@
qu.2.3.part.1.maple=ans:=$I:
LinearAlgebra[Equal](ans,$RESPONSE)@
qu.2.3.part.1.type=maple@
qu.2.3.question=<p>If matrix A is $b<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow></mstyle></math>$b, what is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>A</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>?</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>A</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.2.4.mode=Inline@
qu.2.4.name=Det and Inv of a 3x3 matrix@
qu.2.4.comment=<p>The determinant is $dt.</p>
<p>The inverse is $invpretty.</p>@
qu.2.4.editing=useHTML@
qu.2.4.solution=@
qu.2.4.algorithm=$m=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](3,generator=rand(-3..3),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m1) = 0 do m1 := LinearAlgebra[RandomMatrix](3, generator = rand(-3 .. 3), attributes = [nonsingular]) end do:
m2:=LinearAlgebra[Determinant](m1):
m3:=MathML[ExportPresentation](LinearAlgebra[MatrixInverse](m1)):
m4:=MathML[ExportPresentation](m1):
m5:=LinearAlgebra[MatrixInverse](m1):
m6:=m2*m5:
m7:=MathML[ExportPresentation](m6):
m2,convert(m3,string),m4,convert(m5,string),convert(m6,string), m7
");
$dt=switch(0,$m);
$invpretty=switch(1,$m);
$A=switch(2,$m);
$inv=switch(3,$m);
$adj=switch(4,$m);
$adjpretty=switch(5,$m);@
qu.2.4.uid=0edaedba-9887-4cf2-bffd-d2f44d76c593@
qu.2.4.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants And Inverses;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Students Use Equation Editor;
@
qu.2.4.weighting=1,1@
qu.2.4.numbering=alpha@
qu.2.4.part.1.name=sro_id_1@
qu.2.4.part.1.answer.units=@
qu.2.4.part.1.numStyle=   @
qu.2.4.part.1.editing=useHTML@
qu.2.4.part.1.showUnits=false@
qu.2.4.part.1.question=(Unset)@
qu.2.4.part.1.mode=Numeric@
qu.2.4.part.1.grading=exact_value@
qu.2.4.part.1.negStyle=both@
qu.2.4.part.1.answer.num=$dt@
qu.2.4.part.2.name=sro_id_2@
qu.2.4.part.2.maple_answer=printf("$invpretty");@
qu.2.4.part.2.editing=useHTML@
qu.2.4.part.2.question=(Unset)@
qu.2.4.part.2.libname=@
qu.2.4.part.2.mode=Maple@
qu.2.4.part.2.allow2d=2@
qu.2.4.part.2.plot=@
qu.2.4.part.2.maple=ans:=$inv:
grade:=0:
for i from 1 to 3 do
for j from 1 to 3 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.11112:
end if;
end;
end;
grade;@
qu.2.4.part.2.type=maple@
qu.2.4.question=<p>What is the determinant of the following matrix?</p><p>A=$A</p><p>&nbsp;</p><p>det(A)=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the inverse of A?</p><p>(To enter your answer, right-click on the equation editor, select&nbsp; <img alt="" width="60" height="45" src="__BASE_URI__images/equationeditor2.PNG" /> and then <img alt="" width="75" height="79" src="__BASE_URI__images/equationeditor3.PNG" /> to bring up an empty matrix. Next, replace each letter with your numeric answer, tabbing between cells.)</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;<span>Note: Please enter the values <em>exactly</em> - ie. as fractions or whole numbers, not as decimal numbers.</span></p><p>&nbsp;</p>@

qu.2.5.mode=Inline@
qu.2.5.name=Det and Inv of a 2x2 matrix, with A*A inverse question@
qu.2.5.comment=<p>The determinant is $det.</p>
<p>The inverse is $invpretty.</p>@
qu.2.5.editing=useHTML@
qu.2.5.solution=@
qu.2.5.algorithm=$m=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](2,generator=rand(-9..9),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m1) = 0 do m1 := LinearAlgebra[RandomMatrix](2, generator = rand(-9 .. 9), attributes = [nonsingular]) end do:
m2:=LinearAlgebra[MatrixInverse](m1):
m3:=MathML[ExportPresentation](m2):
m4:=MathML[ExportPresentation](m1):
m5:=LinearAlgebra[Determinant](m1):
m6:=Matrix([[1,0],[0,1]]):
m7:=MathML[ExportPresentation](m6):
convert(m2,string),m3,convert(m4,string),m5,convert(m6,string),convert(m7,string)
");
$inv=switch(0,$m);
$invpretty=switch(1,$m);
$A=switch(2,$m);
$det=switch(3,$m);
$I=switch(4,$m);
$Ipretty=switch(5,$m);@
qu.2.5.uid=87e964e9-3c1c-44ac-85b2-584a9486b29a@
qu.2.5.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants And Inverses, And A*Ainverse;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Students Use Equation Editor;
@
qu.2.5.weighting=1,1,1@
qu.2.5.numbering=alpha@
qu.2.5.part.1.name=sro_id_1@
qu.2.5.part.1.answer.units=@
qu.2.5.part.1.numStyle=   @
qu.2.5.part.1.editing=useHTML@
qu.2.5.part.1.showUnits=false@
qu.2.5.part.1.question=(Unset)@
qu.2.5.part.1.mode=Numeric@
qu.2.5.part.1.grading=exact_value@
qu.2.5.part.1.negStyle=both@
qu.2.5.part.1.answer.num=$det@
qu.2.5.part.2.name=sro_id_2@
qu.2.5.part.2.maple_answer=printf("$invpretty");@
qu.2.5.part.2.editing=useHTML@
qu.2.5.part.2.question=(Unset)@
qu.2.5.part.2.libname=@
qu.2.5.part.2.mode=Maple@
qu.2.5.part.2.allow2d=2@
qu.2.5.part.2.plot=@
qu.2.5.part.2.maple=ans:=$inv:
grade:=0:
for i from 1 to 2 do
for j from 1 to 2 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.25:
end if;
end;
end;
grade;@
qu.2.5.part.2.type=maple@
qu.2.5.part.3.name=sro_id_3@
qu.2.5.part.3.maple_answer=printf("$Ipretty");@
qu.2.5.part.3.editing=useHTML@
qu.2.5.part.3.question=(Unset)@
qu.2.5.part.3.libname=@
qu.2.5.part.3.mode=Maple@
qu.2.5.part.3.allow2d=2@
qu.2.5.part.3.plot=@
qu.2.5.part.3.maple=ans:=$I:
grade:=0:
for i from 1 to 2 do
for j from 1 to 2 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.25:
end if;
end;
end;
grade;@
qu.2.5.part.3.type=maple@
qu.2.5.question=<p>What is the determinant of the following matrix?</p><p>A=$A</p><p>&nbsp;</p><p>det(A)=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the inverse of A?</p><p>&nbsp;(To input your answer, right-click on the box below to bring up the symbols bar, select the button with a square made out of 9 smaller squares and select the appropriate size of matrix. If the correct size is not shown, select <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mi>m</mi></mrow></mstyle></math>and set the dimensions yourself. )</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>A</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math></p><p>&nbsp;</p><p>&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span></p><p>&nbsp;<span>Note: Please enter the values <em>exactly</em> - ie. as fractions or whole numbers, not as decimal numbers.</span></p><p>&nbsp;</p>@

qu.2.6.mode=Inline@
qu.2.6.name=Det and Adj of a 3x3 matrix@
qu.2.6.comment=<p>The determinant is $dt.</p>
<p>The adjoint matrix is $adjpretty.</p>@
qu.2.6.editing=useHTML@
qu.2.6.solution=@
qu.2.6.algorithm=$m=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](3,generator=rand(-3..3),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m1) = 0 do m1 := LinearAlgebra[RandomMatrix](3, generator = rand(-3 .. 3), attributes = [nonsingular]) end do:
m2:=LinearAlgebra[Determinant](m1):
m3:=MathML[ExportPresentation](LinearAlgebra[MatrixInverse](m1)):
m4:=MathML[ExportPresentation](m1):
m5:=LinearAlgebra[MatrixInverse](m1):
m6:=m2*m5:
m7:=MathML[ExportPresentation](m6):
m2,m3,m4,convert(m5,string),convert(m6,string), m7
");
$dt=switch(0,$m);
$inv=switch(1,$m);
$A=switch(2,$m);
$minv=switch(3,$m);
$adj=switch(4,$m);
$adjpretty=switch(5,$m);@
qu.2.6.uid=e42d80c4-2076-478e-b37e-bf9a1f14dbfa@
qu.2.6.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants And Inverses;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Students Use Equation Editor;
@
qu.2.6.weighting=1,1@
qu.2.6.numbering=alpha@
qu.2.6.part.1.name=sro_id_1@
qu.2.6.part.1.answer.units=@
qu.2.6.part.1.numStyle=   @
qu.2.6.part.1.editing=useHTML@
qu.2.6.part.1.showUnits=false@
qu.2.6.part.1.question=(Unset)@
qu.2.6.part.1.mode=Numeric@
qu.2.6.part.1.grading=exact_value@
qu.2.6.part.1.negStyle=both@
qu.2.6.part.1.answer.num=$dt@
qu.2.6.part.2.name=sro_id_2@
qu.2.6.part.2.maple_answer=printf("$adjpretty")@
qu.2.6.part.2.editing=useHTML@
qu.2.6.part.2.question=(Unset)@
qu.2.6.part.2.libname=@
qu.2.6.part.2.mode=Maple@
qu.2.6.part.2.allow2d=2@
qu.2.6.part.2.plot=@
qu.2.6.part.2.maple=ans:=$adj:
grade:=0:
for i from 1 to 3 do
for j from 1 to 3 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.1112:
end if;
end;
end;
grade;@
qu.2.6.part.2.type=maple@
qu.2.6.question=<p>What is the determinant of the following matrix?</p><p>$A</p><p>det(A)=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the adjoint matrix of A?</p><p>(To enter your answer, right-click on the equation editor, select&nbsp; <img alt="" width="60" height="45" src="__BASE_URI__images/equationeditor2.PNG" /> and then <img alt="" width="75" height="79" src="__BASE_URI__images/equationeditor3.PNG" /> to bring up an empty matrix. Next, replace each letter with your numeric answer, tabbing between cells.)</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p><span><span>Note: Please enter the values <em>exactly</em> - ie. as fractions or whole numbers, not as decimal numbers.</span></span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.2.7.mode=Inline@
qu.2.7.name=Det and Inv of a 2x2 matrix@
qu.2.7.comment=<p>The determinant is $det.</p>
<p>The inverse is $invpretty.</p>@
qu.2.7.editing=useHTML@
qu.2.7.solution=@
qu.2.7.algorithm=$m=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](2,generator=rand(-9..9),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m1) = 0 do m1 := LinearAlgebra[RandomMatrix](2, generator = rand(-9 .. 9), attributes = [nonsingular]) end do:
m2:=LinearAlgebra[MatrixInverse](m1):
m3:=MathML[ExportPresentation](m2):
m4:=MathML[ExportPresentation](m1):
m5:=LinearAlgebra[Determinant](m1):
convert(m2,string),m3,convert(m4,string),m5
");
$inv=switch(0,$m);
$invpretty=switch(1,$m);
$A=switch(2,$m);
$det=switch(3,$m);@
qu.2.7.uid=4eecfc02-81b2-4443-9ca4-923c5917966d@
qu.2.7.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants And Inverses;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Students Use Equation Editor;
@
qu.2.7.weighting=1,1@
qu.2.7.numbering=alpha@
qu.2.7.part.1.name=sro_id_1@
qu.2.7.part.1.answer.units=@
qu.2.7.part.1.numStyle=   @
qu.2.7.part.1.editing=useHTML@
qu.2.7.part.1.showUnits=false@
qu.2.7.part.1.question=(Unset)@
qu.2.7.part.1.mode=Numeric@
qu.2.7.part.1.grading=exact_value@
qu.2.7.part.1.negStyle=both@
qu.2.7.part.1.answer.num=$det@
qu.2.7.part.2.name=sro_id_2@
qu.2.7.part.2.maple_answer=printf("$invpretty");@
qu.2.7.part.2.editing=useHTML@
qu.2.7.part.2.question=(Unset)@
qu.2.7.part.2.libname=@
qu.2.7.part.2.mode=Maple@
qu.2.7.part.2.allow2d=2@
qu.2.7.part.2.plot=@
qu.2.7.part.2.maple=ans:=$inv:
grade:=0:
for i from 1 to 2 do
for j from 1 to 2 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.25:
end if;
end;
end;
grade;@
qu.2.7.part.2.type=maple@
qu.2.7.question=<p>What is the determinant of the following matrix?</p><p>$A</p><p>det(A)=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the inverse of A?</p><p>&nbsp;</p><p>(To input your answer, right-click on the box below to bring up the symbols bar, select the button with a square made out of 9 smaller squares and select the appropriate size of matrix. If the correct size is not shown, select <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mi>m</mi></mrow></mstyle></math>and set the dimensions yourself. )</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p><span>Note:&nbsp; Please enter the values <em>exactly</em> - ie. as fractions or whole numbers, not&nbsp; as decimal numbers.</span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.2.8.mode=Inline@
qu.2.8.name=Det and Inv of a 4x4 matrix@
qu.2.8.comment=<p>The determinant is $dt.</p>
<p>The inverse is $invpretty.</p>@
qu.2.8.editing=useHTML@
qu.2.8.solution=@
qu.2.8.algorithm=$m=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](4,generator=rand(-3..3),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m1) = 0 do m1 := LinearAlgebra[RandomMatrix](4, generator = rand(-3 .. 3), attributes = [nonsingular]) end do:
m2:=LinearAlgebra[Determinant](m1):
m3:=MathML[ExportPresentation](LinearAlgebra[MatrixInverse](m1)):
m4:=MathML[ExportPresentation](m1):
m5:=LinearAlgebra[MatrixInverse](m1):
m6:=m2*m5:
m7:=MathML[ExportPresentation](m6):
m2,m3,m4,convert(m5,string),convert(m6,string), m7
");
$dt=switch(0,$m);
$invpretty=switch(1,$m);
$A=switch(2,$m);
$inv=switch(3,$m);
$adj=switch(4,$m);
$adjpretty=switch(5,$m);@
qu.2.8.uid=8476d7b2-7769-45ed-995f-8e0d16fb4297@
qu.2.8.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants And Inverses;
  Author=Katherine Dare;
  Difficulty=Hard;
  Feature=Students Use Equation Editor;
@
qu.2.8.weighting=1,1@
qu.2.8.numbering=alpha@
qu.2.8.part.1.name=sro_id_1@
qu.2.8.part.1.answer.units=@
qu.2.8.part.1.numStyle=   @
qu.2.8.part.1.editing=useHTML@
qu.2.8.part.1.showUnits=false@
qu.2.8.part.1.question=(Unset)@
qu.2.8.part.1.mode=Numeric@
qu.2.8.part.1.grading=exact_value@
qu.2.8.part.1.negStyle=both@
qu.2.8.part.1.answer.num=$dt@
qu.2.8.part.2.name=sro_id_2@
qu.2.8.part.2.maple_answer=printf("$invpretty");@
qu.2.8.part.2.editing=useHTML@
qu.2.8.part.2.question=(Unset)@
qu.2.8.part.2.libname=@
qu.2.8.part.2.mode=Maple@
qu.2.8.part.2.allow2d=2@
qu.2.8.part.2.plot=@
qu.2.8.part.2.maple=ans:=$inv:
grade:=0:
for i from 1 to 4 do
for j from 1 to 4 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.0625:
end if;
end;
end;
grade;@
qu.2.8.part.2.type=maple@
qu.2.8.question=<p>What is the determinant of the following matrix?</p><p>A=$A</p><p>&nbsp;</p><p>det(A)=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the inverse of A?</p><p>(To enter your answer, right-click on the equation editor, select&nbsp; <img alt="" width="60" height="45" src="__BASE_URI__images/equationeditor2.PNG" />, then&nbsp;<img alt="" width="85" height="45" src="__BASE_URI__images/equationeditor4.PNG" /> and then choose 4x4 on this screen: <img alt="" width="75" height="55" src="__BASE_URI__images/equationeditor5.PNG" />to bring up an empty matrix. Next, replace each letter with your numeric answer, tabbing between cells.)</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;<span>Note: Please enter the values <em>exactly</em> - ie. as fractions or whole numbers, not as decimal numbers.</span></p><p>&nbsp;</p>@

qu.2.9.mode=Inline@
qu.2.9.name=Det and Adj of a 4x4 matrix@
qu.2.9.comment=<p>The determinant is $dt.</p>
<p>The adjunct matrix is $adjpretty.</p>@
qu.2.9.editing=useHTML@
qu.2.9.solution=@
qu.2.9.algorithm=$m=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](4,generator=rand(-3..3),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m1) = 0 do m1 := LinearAlgebra[RandomMatrix](4, generator = rand(-3 .. 3), attributes = [nonsingular]) end do:
m2:=LinearAlgebra[Determinant](m1):
m3:=MathML[ExportPresentation](LinearAlgebra[MatrixInverse](m1)):
m4:=MathML[ExportPresentation](m1):
m5:=LinearAlgebra[MatrixInverse](m1):
m6:=m2*m5:
m7:=MathML[ExportPresentation](m6):
m2,m3,m4,convert(m5,string),convert(m6,string), m7
");
$dt=switch(0,$m);
$inv=switch(1,$m);
$A=switch(2,$m);
$minv=switch(3,$m);
$adj=switch(4,$m);
$adjpretty=switch(5,$m);@
qu.2.9.uid=99a06c24-5ff2-4a1b-bc8d-af740e39470e@
qu.2.9.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants And Inverses;
  Author=Katherine Dare;
  Difficulty=Hard;
  Feature=Students Use Equation Editor;
@
qu.2.9.weighting=1,1@
qu.2.9.numbering=alpha@
qu.2.9.part.1.name=sro_id_1@
qu.2.9.part.1.answer.units=@
qu.2.9.part.1.numStyle=   @
qu.2.9.part.1.editing=useHTML@
qu.2.9.part.1.showUnits=false@
qu.2.9.part.1.question=(Unset)@
qu.2.9.part.1.mode=Numeric@
qu.2.9.part.1.grading=exact_value@
qu.2.9.part.1.negStyle=both@
qu.2.9.part.1.answer.num=$dt@
qu.2.9.part.2.name=sro_id_2@
qu.2.9.part.2.maple_answer=printf("$adjpretty")@
qu.2.9.part.2.editing=useHTML@
qu.2.9.part.2.question=(Unset)@
qu.2.9.part.2.libname=@
qu.2.9.part.2.mode=Maple@
qu.2.9.part.2.allow2d=2@
qu.2.9.part.2.plot=@
qu.2.9.part.2.maple=ans:=$adj:
grade:=0:
for i from 1 to 4 do
for j from 1 to 4 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.0625:
end if;
end;
end;
grade;@
qu.2.9.part.2.type=maple@
qu.2.9.question=<p>What is the determinant of the following matrix?</p><p>$A</p><p>det(A)=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the adjoint matrix of A?</p><p>(To enter your answer, right-click on the equation editor, select&nbsp; <img alt="" width="60" height="45" src="__BASE_URI__images/equationeditor2.PNG" />, then&nbsp;<img alt="" width="85" height="45" src="__BASE_URI__images/equationeditor4.PNG" /> and then choose 4x4 on this screen: <img alt="" width="75" height="55" src="__BASE_URI__images/equationeditor5.PNG" />to bring up an empty matrix. Next, replace each letter with your numeric answer, tabbing between cells.)</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;<span>Note: Please enter the values <em>exactly</em> - ie. as fractions or whole numbers, not as decimal numbers.</span></p><p>&nbsp;</p>@

qu.3.topic=Determinants@

qu.3.1.mode=Inline@
qu.3.1.name=Determinants of Related Matrices - product of two square matrices@
qu.3.1.comment=<p>The determinant of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>is $detA.</p>
<p>The determinant of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>is $detB.</p>
<p>Therefore the determinant of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>AB</mi></mrow></mstyle></math> is $detAB.</p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$v=maple("
randomize():
with (LinearAlgebra):
v1:=RandomMatrix(2,generator=rand(-5..5)):
v2:=Determinant(v1):
v3:=MathML[ExportPresentation](v1):
v4:=RandomMatrix(2,generator=rand(-5..5)):
v5:=Determinant(v4):
v6:=MathML[ExportPresentation](v4):
v2,convert(v3,string),v5,convert(v6,string)
");
$detA=switch(0,$v);
$A=switch(1,$v);
$detB=switch(2,$v);
$B=switch(3,$v);
$detAB=$detA*$detB;@
qu.3.1.uid=7af4fc93-c7fc-4e89-a3fd-c526d05963c1@
qu.3.1.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants Of Related Matrices;
  Author=Katherine Dare;
  Difficulty=Easy;
@
qu.3.1.weighting=1,1,1@
qu.3.1.numbering=alpha@
qu.3.1.part.1.name=sro_id_1@
qu.3.1.part.1.answer.units=@
qu.3.1.part.1.numStyle=   @
qu.3.1.part.1.editing=useHTML@
qu.3.1.part.1.showUnits=false@
qu.3.1.part.1.question=(Unset)@
qu.3.1.part.1.mode=Numeric@
qu.3.1.part.1.grading=exact_value@
qu.3.1.part.1.negStyle=both@
qu.3.1.part.1.answer.num=$detA@
qu.3.1.part.2.name=sro_id_2@
qu.3.1.part.2.answer.units=@
qu.3.1.part.2.numStyle=   @
qu.3.1.part.2.editing=useHTML@
qu.3.1.part.2.showUnits=false@
qu.3.1.part.2.question=(Unset)@
qu.3.1.part.2.mode=Numeric@
qu.3.1.part.2.grading=exact_value@
qu.3.1.part.2.negStyle=both@
qu.3.1.part.2.answer.num=$detB@
qu.3.1.part.3.name=sro_id_3@
qu.3.1.part.3.answer.units=@
qu.3.1.part.3.numStyle=   @
qu.3.1.part.3.editing=useHTML@
qu.3.1.part.3.showUnits=false@
qu.3.1.part.3.question=(Unset)@
qu.3.1.part.3.mode=Numeric@
qu.3.1.part.3.grading=exact_value@
qu.3.1.part.3.negStyle=both@
qu.3.1.part.3.answer.num=$detAB@
qu.3.1.question=<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>=$A</p><p>and</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>=$B,</p><p>&nbsp;</p><p>What is the determinant of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>A</mi></mrow></mfenced></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the determinant of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>B</mi></mrow></mfenced></mrow></mstyle></math>=<span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the determinant of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>AB</mi></mrow></mstyle></math>?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>AB</mi></mrow></mfenced></mrow></mstyle></math>=<span>&nbsp;</span><3><span>&nbsp;</span></p>@

qu.3.2.mode=Inline@
qu.3.2.name=Det of a 3x3 matrix, Invertibility@
qu.3.2.comment=<p>The determinant of $A is $det.</p>
<p>Because the determinant $feedback2, A $feedback invertible.</p>@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$m=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](3,generator=rand(-1..2)):
m2:=LinearAlgebra[Determinant](m1):
m4:=MathML[ExportPresentation](m1):
if m2=0 then m3:=0:
else m3:=1:
end if:
m2,convert(m4,string),m3
");
$det=switch(0,$m);
$A=switch(1,$m);
$k=switch(2,$m);
$ANSWER=switch($k,'No','Yes');
$WRONG=switch($k,'Yes','No');
$feedback=switch($k,'is not','is');
$feedback2=switch($k,'is equal to zero','is not equal to zero');@
qu.3.2.uid=82178e97-71e5-4f2a-9bcc-61f8307409d0@
qu.3.2.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants, Invertibility;
  Author=Katherine Dare;
  Difficulty=Easy;
@
qu.3.2.weighting=1,1@
qu.3.2.numbering=alpha@
qu.3.2.part.1.name=sro_id_1@
qu.3.2.part.1.answer.units=@
qu.3.2.part.1.numStyle=   @
qu.3.2.part.1.editing=useHTML@
qu.3.2.part.1.showUnits=false@
qu.3.2.part.1.question=(Unset)@
qu.3.2.part.1.mode=Numeric@
qu.3.2.part.1.grading=exact_value@
qu.3.2.part.1.negStyle=both@
qu.3.2.part.1.answer.num=$det@
qu.3.2.part.2.grader=exact@
qu.3.2.part.2.name=sro_id_2@
qu.3.2.part.2.editing=useHTML@
qu.3.2.part.2.display.permute=true@
qu.3.2.part.2.answer.3=Not Enough Info@
qu.3.2.part.2.question=(Unset)@
qu.3.2.part.2.answer.2=$WRONG@
qu.3.2.part.2.answer.1=$ANSWER@
qu.3.2.part.2.mode=List@
qu.3.2.part.2.display=menu@
qu.3.2.part.2.credit.3=0.0@
qu.3.2.part.2.credit.2=0.0@
qu.3.2.part.2.credit.1=1.0@
qu.3.2.question=<p>What is the determinant of the following matrix?</p><p>A=$A</p><p>det(A)=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>Is A invertible?</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p>@

qu.3.3.mode=Inline@
qu.3.3.name=Det of a 5x5 matrix@
qu.3.3.comment=<p>The determinant is $dt.</p>@
qu.3.3.editing=useHTML@
qu.3.3.solution=@
qu.3.3.algorithm=$m=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](5,generator=rand(-3..3)):
m2:=LinearAlgebra[Determinant](m1):
m3:=MathML[ExportPresentation](m1):
m2,m3
");
$dt=switch(0,$m);
$A=switch(1,$m);@
qu.3.3.uid=b9bb1a98-80ef-420e-be1d-f19bc747236d@
qu.3.3.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants;
  Author=Katherine Dare;
  Difficulty=Hard;
@
qu.3.3.weighting=1@
qu.3.3.numbering=alpha@
qu.3.3.part.1.name=sro_id_1@
qu.3.3.part.1.answer.units=@
qu.3.3.part.1.numStyle=   @
qu.3.3.part.1.editing=useHTML@
qu.3.3.part.1.showUnits=false@
qu.3.3.part.1.question=(Unset)@
qu.3.3.part.1.mode=Numeric@
qu.3.3.part.1.grading=exact_value@
qu.3.3.part.1.negStyle=both@
qu.3.3.part.1.answer.num=$dt@
qu.3.3.question=<p>What is the determinant of the following matrix?</p><p>A=$A</p><p>det(A)=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.3.4.mode=Inline@
qu.3.4.name=Determinants of Related Matrices - every element multiplied by a scalar@
qu.3.4.comment=<p>B is A, but with every element multiplied by $c. Therefore, the determinant of B is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$c</mi><mrow><mn>3</mn></mrow></msup></mrow></mstyle></math>det(A)=$ans. Note that if the matrix was <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mi mathvariant='normal'>n</mi></mrow></mstyle></math>, the determinant would be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$c</mi><mrow><mi mathvariant='normal'>n</mi></mrow></msup></mrow></mstyle></math>det(A).</p>@
qu.3.4.editing=useHTML@
qu.3.4.solution=@
qu.3.4.algorithm=$det=range(2,10);
$c=range(2,4);
$ans=$c^3*$det;
$v=maple("
A := MathML[ExportPresentation](Matrix([[a, d, g], [b, e, h],[c,f,i]])):
v1 := MathML[ExportPresentation](Matrix([[$c*a, $c*d, $c*g], [$c*b, $c*e, $c*h],[$c*c,$c*f,$c*i]])):
convert(A,string),convert(v1,string)
");
$A=switch(0,$v);
$B=switch(1,$v);@
qu.3.4.uid=d0120bce-eb4f-4d03-9975-33de24e05d6d@
qu.3.4.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants Of Related Matrices - Every Element Multiplied By A Scalar;
  Author=Katherine Dare;
  Difficulty=Easy;
@
qu.3.4.weighting=1@
qu.3.4.numbering=alpha@
qu.3.4.part.1.name=sro_id_1@
qu.3.4.part.1.answer.units=@
qu.3.4.part.1.numStyle=   arithmetic@
qu.3.4.part.1.editing=useHTML@
qu.3.4.part.1.showUnits=false@
qu.3.4.part.1.err=0.05@
qu.3.4.part.1.question=(Unset)@
qu.3.4.part.1.mode=Numeric@
qu.3.4.part.1.grading=toler_abs@
qu.3.4.part.1.negStyle=minus@
qu.3.4.part.1.answer.num=$ans@
qu.3.4.question=<p>If the determinant of A=$A is $det, what is the determinant of B=$B?</p><p>&nbsp;</p><p>det(B)=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.3.5.mode=Inline@
qu.3.5.name=Det of a 3x3 matrix@
qu.3.5.comment=<p>The determinant of $A is $det.</p>@
qu.3.5.editing=useHTML@
qu.3.5.solution=@
qu.3.5.algorithm=$m=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](3,generator=rand(-5..5),attributes=[nonsingular]):
m2:=LinearAlgebra[Determinant](m1):
m4:=MathML[ExportPresentation](m1):
m2,convert(m4,string)
");
$det=switch(0,$m);
$A=switch(1,$m);@
qu.3.5.uid=259f4479-a667-48ad-b561-be31f39d12f6@
qu.3.5.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants;
  Author=Katherine Dare;
  Difficulty=Medium;
@
qu.3.5.weighting=1@
qu.3.5.numbering=alpha@
qu.3.5.part.1.name=sro_id_1@
qu.3.5.part.1.answer.units=@
qu.3.5.part.1.numStyle=   @
qu.3.5.part.1.editing=useHTML@
qu.3.5.part.1.showUnits=false@
qu.3.5.part.1.question=(Unset)@
qu.3.5.part.1.mode=Numeric@
qu.3.5.part.1.grading=exact_value@
qu.3.5.part.1.negStyle=both@
qu.3.5.part.1.answer.num=$det@
qu.3.5.question=<p>What is the determinant of the following matrix?</p><p>$A</p><p>det(A)=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.3.6.mode=Inline@
qu.3.6.name=Det and Inv of a 3x3 matrix@
qu.3.6.comment=<p>The determinant is $dt.</p>
<p>The inverse is $invpretty.</p>@
qu.3.6.editing=useHTML@
qu.3.6.solution=@
qu.3.6.algorithm=$m=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](3,generator=rand(-3..3),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m1) = 0 do m1 := LinearAlgebra[RandomMatrix](3, generator = rand(-3 .. 3), attributes = [nonsingular]) end do:
m2:=LinearAlgebra[Determinant](m1):
m3:=MathML[ExportPresentation](LinearAlgebra[MatrixInverse](m1)):
m4:=MathML[ExportPresentation](m1):
m5:=LinearAlgebra[MatrixInverse](m1):
m6:=m2*m5:
m7:=MathML[ExportPresentation](m6):
m2,convert(m3,string),m4,convert(m5,string),convert(m6,string), m7
");
$dt=switch(0,$m);
$invpretty=switch(1,$m);
$A=switch(2,$m);
$inv=switch(3,$m);
$adj=switch(4,$m);
$adjpretty=switch(5,$m);@
qu.3.6.uid=0edaedba-9887-4cf2-bffd-d2f44d76c593@
qu.3.6.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants And Inverses;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Students Use Equation Editor;
@
qu.3.6.weighting=1,1@
qu.3.6.numbering=alpha@
qu.3.6.part.1.name=sro_id_1@
qu.3.6.part.1.answer.units=@
qu.3.6.part.1.numStyle=   @
qu.3.6.part.1.editing=useHTML@
qu.3.6.part.1.showUnits=false@
qu.3.6.part.1.question=(Unset)@
qu.3.6.part.1.mode=Numeric@
qu.3.6.part.1.grading=exact_value@
qu.3.6.part.1.negStyle=both@
qu.3.6.part.1.answer.num=$dt@
qu.3.6.part.2.name=sro_id_2@
qu.3.6.part.2.maple_answer=printf("$invpretty");@
qu.3.6.part.2.editing=useHTML@
qu.3.6.part.2.question=(Unset)@
qu.3.6.part.2.libname=@
qu.3.6.part.2.mode=Maple@
qu.3.6.part.2.allow2d=2@
qu.3.6.part.2.plot=@
qu.3.6.part.2.maple=ans:=$inv:
grade:=0:
for i from 1 to 3 do
for j from 1 to 3 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.11112:
end if;
end;
end;
grade;@
qu.3.6.part.2.type=maple@
qu.3.6.question=<p>What is the determinant of the following matrix?</p><p>A=$A</p><p>&nbsp;</p><p>det(A)=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the inverse of A?</p><p>(To enter your answer, right-click on the equation editor, select&nbsp; <img alt="" width="60" height="45" src="__BASE_URI__images/equationeditor2.PNG" /> and then <img alt="" width="75" height="79" src="__BASE_URI__images/equationeditor3.PNG" /> to bring up an empty matrix. Next, replace each letter with your numeric answer, tabbing between cells.)</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;<span>Note: Please enter the values <em>exactly</em> - ie. as fractions or whole numbers, not as decimal numbers.</span></p><p>&nbsp;</p>@

qu.3.7.mode=Inline@
qu.3.7.name=Determinants of Related Matrices - row multiplied by a scalar@
qu.3.7.comment=<p>B is A, but with one $feedback multiplied by $c. Therefore, the determinant of B is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$c</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow></mstyle></math>det(A).</p>@
qu.3.7.editing=useHTML@
qu.3.7.solution=@
qu.3.7.algorithm=$det=range(20,50);
$c=range(2,5);
$ans=$det*$c;
$k=range(1,6);
$v=maple("
A := MathML[ExportPresentation](Matrix([[a, d, g], [b, e, h],[c,f,i]])):
v1 := MathML[ExportPresentation](Matrix([[$c*a, $c*d, $c*g], [b, e, h],[c,f,i]])):
v2 := MathML[ExportPresentation](Matrix([[a, d, g], [$c*b, $c*e, $c*h],[c,f,i]])):
v3 := MathML[ExportPresentation](Matrix([[a, d, g], [b, e, h],[$c*c,$c*f,$c*i]])):
v4 := MathML[ExportPresentation](Matrix([[$c*a, d, g], [$c*b, e, h],[$c*c,f,i]])):
v5 := MathML[ExportPresentation](Matrix([[a, $c*d, g], [b, $c*e, h],[c,$c*f,i]])):
v6 := MathML[ExportPresentation](Matrix([[a, d, $c*g], [b, e, $c*h],[c,f,$c*i]])):
convert(A,string),convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(v5,string),convert(v6,string)
");
$A=switch(0,$v);
$B=switch($k,$v);
$feedback=switch($k,'row','row','row','column','column','column');@
qu.3.7.uid=86b6d8cd-35a7-4cf4-b20b-5ac5a75630b0@
qu.3.7.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants Of Related Matrices - Row Multiplied By A Scalar;
  Author=Katherine Dare;
  Difficulty=Easy;
@
qu.3.7.weighting=1@
qu.3.7.numbering=alpha@
qu.3.7.part.1.name=sro_id_1@
qu.3.7.part.1.answer.units=@
qu.3.7.part.1.numStyle=   arithmetic@
qu.3.7.part.1.editing=useHTML@
qu.3.7.part.1.showUnits=false@
qu.3.7.part.1.err=0.05@
qu.3.7.part.1.question=(Unset)@
qu.3.7.part.1.mode=Numeric@
qu.3.7.part.1.grading=toler_abs@
qu.3.7.part.1.negStyle=minus@
qu.3.7.part.1.answer.num=$ans@
qu.3.7.question=<p>If the determinant of A=$A is $det, what is the determinant of B=$B?</p><p>&nbsp;</p><p>det(B)=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.3.8.mode=Inline@
qu.3.8.name=Det and Inv of a 2x2 matrix, with A*A inverse question@
qu.3.8.comment=<p>The determinant is $det.</p>
<p>The inverse is $invpretty.</p>@
qu.3.8.editing=useHTML@
qu.3.8.solution=@
qu.3.8.algorithm=$m=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](2,generator=rand(-9..9),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m1) = 0 do m1 := LinearAlgebra[RandomMatrix](2, generator = rand(-9 .. 9), attributes = [nonsingular]) end do:
m2:=LinearAlgebra[MatrixInverse](m1):
m3:=MathML[ExportPresentation](m2):
m4:=MathML[ExportPresentation](m1):
m5:=LinearAlgebra[Determinant](m1):
m6:=Matrix([[1,0],[0,1]]):
m7:=MathML[ExportPresentation](m6):
convert(m2,string),m3,convert(m4,string),m5,convert(m6,string),convert(m7,string)
");
$inv=switch(0,$m);
$invpretty=switch(1,$m);
$A=switch(2,$m);
$det=switch(3,$m);
$I=switch(4,$m);
$Ipretty=switch(5,$m);@
qu.3.8.uid=87e964e9-3c1c-44ac-85b2-584a9486b29a@
qu.3.8.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants And Inverses, And A*Ainverse;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Students Use Equation Editor;
@
qu.3.8.weighting=1,1,1@
qu.3.8.numbering=alpha@
qu.3.8.part.1.name=sro_id_1@
qu.3.8.part.1.answer.units=@
qu.3.8.part.1.numStyle=   @
qu.3.8.part.1.editing=useHTML@
qu.3.8.part.1.showUnits=false@
qu.3.8.part.1.question=(Unset)@
qu.3.8.part.1.mode=Numeric@
qu.3.8.part.1.grading=exact_value@
qu.3.8.part.1.negStyle=both@
qu.3.8.part.1.answer.num=$det@
qu.3.8.part.2.name=sro_id_2@
qu.3.8.part.2.maple_answer=printf("$invpretty");@
qu.3.8.part.2.editing=useHTML@
qu.3.8.part.2.question=(Unset)@
qu.3.8.part.2.libname=@
qu.3.8.part.2.mode=Maple@
qu.3.8.part.2.allow2d=2@
qu.3.8.part.2.plot=@
qu.3.8.part.2.maple=ans:=$inv:
grade:=0:
for i from 1 to 2 do
for j from 1 to 2 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.25:
end if;
end;
end;
grade;@
qu.3.8.part.2.type=maple@
qu.3.8.part.3.name=sro_id_3@
qu.3.8.part.3.maple_answer=printf("$Ipretty");@
qu.3.8.part.3.editing=useHTML@
qu.3.8.part.3.question=(Unset)@
qu.3.8.part.3.libname=@
qu.3.8.part.3.mode=Maple@
qu.3.8.part.3.allow2d=2@
qu.3.8.part.3.plot=@
qu.3.8.part.3.maple=ans:=$I:
grade:=0:
for i from 1 to 2 do
for j from 1 to 2 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.25:
end if;
end;
end;
grade;@
qu.3.8.part.3.type=maple@
qu.3.8.question=<p>What is the determinant of the following matrix?</p><p>A=$A</p><p>&nbsp;</p><p>det(A)=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the inverse of A?</p><p>&nbsp;(To input your answer, right-click on the box below to bring up the symbols bar, select the button with a square made out of 9 smaller squares and select the appropriate size of matrix. If the correct size is not shown, select <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mi>m</mi></mrow></mstyle></math>and set the dimensions yourself. )</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>A</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math></p><p>&nbsp;</p><p>&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span></p><p>&nbsp;<span>Note: Please enter the values <em>exactly</em> - ie. as fractions or whole numbers, not as decimal numbers.</span></p><p>&nbsp;</p>@

qu.3.9.mode=Inline@
qu.3.9.name=Det and Adj of a 3x3 matrix@
qu.3.9.comment=<p>The determinant is $dt.</p>
<p>The adjoint matrix is $adjpretty.</p>@
qu.3.9.editing=useHTML@
qu.3.9.solution=@
qu.3.9.algorithm=$m=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](3,generator=rand(-3..3),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m1) = 0 do m1 := LinearAlgebra[RandomMatrix](3, generator = rand(-3 .. 3), attributes = [nonsingular]) end do:
m2:=LinearAlgebra[Determinant](m1):
m3:=MathML[ExportPresentation](LinearAlgebra[MatrixInverse](m1)):
m4:=MathML[ExportPresentation](m1):
m5:=LinearAlgebra[MatrixInverse](m1):
m6:=m2*m5:
m7:=MathML[ExportPresentation](m6):
m2,m3,m4,convert(m5,string),convert(m6,string), m7
");
$dt=switch(0,$m);
$inv=switch(1,$m);
$A=switch(2,$m);
$minv=switch(3,$m);
$adj=switch(4,$m);
$adjpretty=switch(5,$m);@
qu.3.9.uid=e42d80c4-2076-478e-b37e-bf9a1f14dbfa@
qu.3.9.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants And Inverses;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Students Use Equation Editor;
@
qu.3.9.weighting=1,1@
qu.3.9.numbering=alpha@
qu.3.9.part.1.name=sro_id_1@
qu.3.9.part.1.answer.units=@
qu.3.9.part.1.numStyle=   @
qu.3.9.part.1.editing=useHTML@
qu.3.9.part.1.showUnits=false@
qu.3.9.part.1.question=(Unset)@
qu.3.9.part.1.mode=Numeric@
qu.3.9.part.1.grading=exact_value@
qu.3.9.part.1.negStyle=both@
qu.3.9.part.1.answer.num=$dt@
qu.3.9.part.2.name=sro_id_2@
qu.3.9.part.2.maple_answer=printf("$adjpretty")@
qu.3.9.part.2.editing=useHTML@
qu.3.9.part.2.question=(Unset)@
qu.3.9.part.2.libname=@
qu.3.9.part.2.mode=Maple@
qu.3.9.part.2.allow2d=2@
qu.3.9.part.2.plot=@
qu.3.9.part.2.maple=ans:=$adj:
grade:=0:
for i from 1 to 3 do
for j from 1 to 3 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.1112:
end if;
end;
end;
grade;@
qu.3.9.part.2.type=maple@
qu.3.9.question=<p>What is the determinant of the following matrix?</p><p>$A</p><p>det(A)=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the adjoint matrix of A?</p><p>(To enter your answer, right-click on the equation editor, select&nbsp; <img alt="" width="60" height="45" src="__BASE_URI__images/equationeditor2.PNG" /> and then <img alt="" width="75" height="79" src="__BASE_URI__images/equationeditor3.PNG" /> to bring up an empty matrix. Next, replace each letter with your numeric answer, tabbing between cells.)</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p><span><span>Note: Please enter the values <em>exactly</em> - ie. as fractions or whole numbers, not as decimal numbers.</span></span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.3.10.mode=Inline@
qu.3.10.name=Det and Inv of a 2x2 matrix@
qu.3.10.comment=<p>The determinant is $det.</p>
<p>The inverse is $invpretty.</p>@
qu.3.10.editing=useHTML@
qu.3.10.solution=@
qu.3.10.algorithm=$m=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](2,generator=rand(-9..9),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m1) = 0 do m1 := LinearAlgebra[RandomMatrix](2, generator = rand(-9 .. 9), attributes = [nonsingular]) end do:
m2:=LinearAlgebra[MatrixInverse](m1):
m3:=MathML[ExportPresentation](m2):
m4:=MathML[ExportPresentation](m1):
m5:=LinearAlgebra[Determinant](m1):
convert(m2,string),m3,convert(m4,string),m5
");
$inv=switch(0,$m);
$invpretty=switch(1,$m);
$A=switch(2,$m);
$det=switch(3,$m);@
qu.3.10.uid=4eecfc02-81b2-4443-9ca4-923c5917966d@
qu.3.10.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants And Inverses;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Students Use Equation Editor;
@
qu.3.10.weighting=1,1@
qu.3.10.numbering=alpha@
qu.3.10.part.1.name=sro_id_1@
qu.3.10.part.1.answer.units=@
qu.3.10.part.1.numStyle=   @
qu.3.10.part.1.editing=useHTML@
qu.3.10.part.1.showUnits=false@
qu.3.10.part.1.question=(Unset)@
qu.3.10.part.1.mode=Numeric@
qu.3.10.part.1.grading=exact_value@
qu.3.10.part.1.negStyle=both@
qu.3.10.part.1.answer.num=$det@
qu.3.10.part.2.name=sro_id_2@
qu.3.10.part.2.maple_answer=printf("$invpretty");@
qu.3.10.part.2.editing=useHTML@
qu.3.10.part.2.question=(Unset)@
qu.3.10.part.2.libname=@
qu.3.10.part.2.mode=Maple@
qu.3.10.part.2.allow2d=2@
qu.3.10.part.2.plot=@
qu.3.10.part.2.maple=ans:=$inv:
grade:=0:
for i from 1 to 2 do
for j from 1 to 2 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.25:
end if;
end;
end;
grade;@
qu.3.10.part.2.type=maple@
qu.3.10.question=<p>What is the determinant of the following matrix?</p><p>$A</p><p>det(A)=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the inverse of A?</p><p>&nbsp;</p><p>(To input your answer, right-click on the box below to bring up the symbols bar, select the button with a square made out of 9 smaller squares and select the appropriate size of matrix. If the correct size is not shown, select <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mi>m</mi></mrow></mstyle></math>and set the dimensions yourself. )</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p><span>Note:&nbsp; Please enter the values <em>exactly</em> - ie. as fractions or whole numbers, not&nbsp; as decimal numbers.</span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.3.11.mode=Inline@
qu.3.11.name=Determinants of Related Matrices - product of two square matrices rule@
qu.3.11.comment=<p>The determinant of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>is $detA.</p>
<p>The determinant of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>is $detB.</p>
<p>Therefore the determinant of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>AB</mi></mrow></mstyle></math> is $detAB.</p>@
qu.3.11.editing=useHTML@
qu.3.11.solution=@
qu.3.11.algorithm=$v=maple("
randomize():
with (LinearAlgebra):
v1:=RandomMatrix(2,generator=rand(-5..5)):
v2:=Determinant(v1):
v3:=MathML[ExportPresentation](v1):
v4:=RandomMatrix(2,generator=rand(-5..5)):
v5:=Determinant(v4):
v6:=MathML[ExportPresentation](v4):
v2,convert(v3,string),v5,convert(v6,string)
");
$detA=switch(0,$v);
$A=switch(1,$v);
$detB=switch(2,$v);
$B=switch(3,$v);
$detAB=$detA*$detB;@
qu.3.11.uid=8d340927-a43f-4f1b-8b21-8271e9071a3e@
qu.3.11.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants Of Related Matrices;
  Author=Katherine Dare;
  Difficulty=Easy;
@
qu.3.11.weighting=1@
qu.3.11.numbering=alpha@
qu.3.11.part.1.name=sro_id_3@
qu.3.11.part.1.answer.units=@
qu.3.11.part.1.numStyle=   @
qu.3.11.part.1.editing=useHTML@
qu.3.11.part.1.showUnits=false@
qu.3.11.part.1.question=(Unset)@
qu.3.11.part.1.mode=Numeric@
qu.3.11.part.1.grading=exact_value@
qu.3.11.part.1.negStyle=both@
qu.3.11.part.1.answer.num=$detAB@
qu.3.11.question=<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi>A</mi></mrow><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow></mstyle></math>=$detA</p><p>and</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi>B</mi></mrow><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow></mstyle></math>=$detB,</p><p>&nbsp;</p><p>What is the determinant of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>AB</mi></mrow></mstyle></math>?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>AB</mi></mrow></mfenced></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.3.12.mode=Inline@
qu.3.12.name=Determinants of Related Matrices - 1 row switched@
qu.3.12.comment=<p>B is A, but with two $feedback switched. Therefore, the determinant of B is -det(A).</p>@
qu.3.12.editing=useHTML@
qu.3.12.solution=@
qu.3.12.algorithm=$det=range(2,10);
$ans=-$det;
$k=range(1,6);
$v=maple("
A := MathML[ExportPresentation](Matrix([[a, d, g], [b, e, h],[c,f,i]])):
v1 := MathML[ExportPresentation](Matrix([[a, d, g], [c, f, i],[b,e,h]])):
v2 := MathML[ExportPresentation](Matrix([[b, e, h], [a, d, g],[c,f,i]])):
v3 := MathML[ExportPresentation](Matrix([[c, f, i], [b, e, h],[a,d,g]])):
v4 := MathML[ExportPresentation](Matrix([[d, a, g], [e, b, h],[f,c,i]])):
v5 := MathML[ExportPresentation](Matrix([[a, g, d], [b, h, e],[c,i,f]])):
v6 := MathML[ExportPresentation](Matrix([[g, d, a], [h, e, b],[i,f,c]])):
convert(A,string),convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(v5,string),convert(v6,string)
");
$A=switch(0,$v);
$B=switch($k,$v);
$feedback=switch($k,'rows','rows','rows','columns','columns','columns');@
qu.3.12.uid=e327bba9-29bf-469e-8f9a-7d717b1bc116@
qu.3.12.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants Of Related Matrices;
  Author=Katherine Dare;
  Difficulty=Easy;
@
qu.3.12.weighting=1@
qu.3.12.numbering=alpha@
qu.3.12.part.1.name=sro_id_1@
qu.3.12.part.1.answer.units=@
qu.3.12.part.1.numStyle=   @
qu.3.12.part.1.editing=useHTML@
qu.3.12.part.1.showUnits=false@
qu.3.12.part.1.question=(Unset)@
qu.3.12.part.1.mode=Numeric@
qu.3.12.part.1.grading=exact_value@
qu.3.12.part.1.negStyle=both@
qu.3.12.part.1.answer.num=$ans@
qu.3.12.question=<p>If the determinant of A=$A is $det, what is the determinant of B=$B?</p><p>&nbsp;</p><p>det(B)=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.3.13.mode=Inline@
qu.3.13.name=Det and Inv of a 4x4 matrix@
qu.3.13.comment=<p>The determinant is $dt.</p>
<p>The inverse is $invpretty.</p>@
qu.3.13.editing=useHTML@
qu.3.13.solution=@
qu.3.13.algorithm=$m=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](4,generator=rand(-3..3),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m1) = 0 do m1 := LinearAlgebra[RandomMatrix](4, generator = rand(-3 .. 3), attributes = [nonsingular]) end do:
m2:=LinearAlgebra[Determinant](m1):
m3:=MathML[ExportPresentation](LinearAlgebra[MatrixInverse](m1)):
m4:=MathML[ExportPresentation](m1):
m5:=LinearAlgebra[MatrixInverse](m1):
m6:=m2*m5:
m7:=MathML[ExportPresentation](m6):
m2,m3,m4,convert(m5,string),convert(m6,string), m7
");
$dt=switch(0,$m);
$invpretty=switch(1,$m);
$A=switch(2,$m);
$inv=switch(3,$m);
$adj=switch(4,$m);
$adjpretty=switch(5,$m);@
qu.3.13.uid=8476d7b2-7769-45ed-995f-8e0d16fb4297@
qu.3.13.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants And Inverses;
  Author=Katherine Dare;
  Difficulty=Hard;
  Feature=Students Use Equation Editor;
@
qu.3.13.weighting=1,1@
qu.3.13.numbering=alpha@
qu.3.13.part.1.name=sro_id_1@
qu.3.13.part.1.answer.units=@
qu.3.13.part.1.numStyle=   @
qu.3.13.part.1.editing=useHTML@
qu.3.13.part.1.showUnits=false@
qu.3.13.part.1.question=(Unset)@
qu.3.13.part.1.mode=Numeric@
qu.3.13.part.1.grading=exact_value@
qu.3.13.part.1.negStyle=both@
qu.3.13.part.1.answer.num=$dt@
qu.3.13.part.2.name=sro_id_2@
qu.3.13.part.2.maple_answer=printf("$invpretty");@
qu.3.13.part.2.editing=useHTML@
qu.3.13.part.2.question=(Unset)@
qu.3.13.part.2.libname=@
qu.3.13.part.2.mode=Maple@
qu.3.13.part.2.allow2d=2@
qu.3.13.part.2.plot=@
qu.3.13.part.2.maple=ans:=$inv:
grade:=0:
for i from 1 to 4 do
for j from 1 to 4 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.0625:
end if;
end;
end;
grade;@
qu.3.13.part.2.type=maple@
qu.3.13.question=<p>What is the determinant of the following matrix?</p><p>A=$A</p><p>&nbsp;</p><p>det(A)=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the inverse of A?</p><p>(To enter your answer, right-click on the equation editor, select&nbsp; <img alt="" width="60" height="45" src="__BASE_URI__images/equationeditor2.PNG" />, then&nbsp;<img alt="" width="85" height="45" src="__BASE_URI__images/equationeditor4.PNG" /> and then choose 4x4 on this screen: <img alt="" width="75" height="55" src="__BASE_URI__images/equationeditor5.PNG" />to bring up an empty matrix. Next, replace each letter with your numeric answer, tabbing between cells.)</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;<span>Note: Please enter the values <em>exactly</em> - ie. as fractions or whole numbers, not as decimal numbers.</span></p><p>&nbsp;</p>@

qu.3.14.mode=Inline@
qu.3.14.name=Det and Adj of a 4x4 matrix@
qu.3.14.comment=<p>The determinant is $dt.</p>
<p>The adjunct matrix is $adjpretty.</p>@
qu.3.14.editing=useHTML@
qu.3.14.solution=@
qu.3.14.algorithm=$m=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](4,generator=rand(-3..3),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m1) = 0 do m1 := LinearAlgebra[RandomMatrix](4, generator = rand(-3 .. 3), attributes = [nonsingular]) end do:
m2:=LinearAlgebra[Determinant](m1):
m3:=MathML[ExportPresentation](LinearAlgebra[MatrixInverse](m1)):
m4:=MathML[ExportPresentation](m1):
m5:=LinearAlgebra[MatrixInverse](m1):
m6:=m2*m5:
m7:=MathML[ExportPresentation](m6):
m2,m3,m4,convert(m5,string),convert(m6,string), m7
");
$dt=switch(0,$m);
$inv=switch(1,$m);
$A=switch(2,$m);
$minv=switch(3,$m);
$adj=switch(4,$m);
$adjpretty=switch(5,$m);@
qu.3.14.uid=99a06c24-5ff2-4a1b-bc8d-af740e39470e@
qu.3.14.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants And Inverses;
  Author=Katherine Dare;
  Difficulty=Hard;
  Feature=Students Use Equation Editor;
@
qu.3.14.weighting=1,1@
qu.3.14.numbering=alpha@
qu.3.14.part.1.name=sro_id_1@
qu.3.14.part.1.answer.units=@
qu.3.14.part.1.numStyle=   @
qu.3.14.part.1.editing=useHTML@
qu.3.14.part.1.showUnits=false@
qu.3.14.part.1.question=(Unset)@
qu.3.14.part.1.mode=Numeric@
qu.3.14.part.1.grading=exact_value@
qu.3.14.part.1.negStyle=both@
qu.3.14.part.1.answer.num=$dt@
qu.3.14.part.2.name=sro_id_2@
qu.3.14.part.2.maple_answer=printf("$adjpretty")@
qu.3.14.part.2.editing=useHTML@
qu.3.14.part.2.question=(Unset)@
qu.3.14.part.2.libname=@
qu.3.14.part.2.mode=Maple@
qu.3.14.part.2.allow2d=2@
qu.3.14.part.2.plot=@
qu.3.14.part.2.maple=ans:=$adj:
grade:=0:
for i from 1 to 4 do
for j from 1 to 4 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.0625:
end if;
end;
end;
grade;@
qu.3.14.part.2.type=maple@
qu.3.14.question=<p>What is the determinant of the following matrix?</p><p>$A</p><p>det(A)=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What is the adjoint matrix of A?</p><p>(To enter your answer, right-click on the equation editor, select&nbsp; <img alt="" width="60" height="45" src="__BASE_URI__images/equationeditor2.PNG" />, then&nbsp;<img alt="" width="85" height="45" src="__BASE_URI__images/equationeditor4.PNG" /> and then choose 4x4 on this screen: <img alt="" width="75" height="55" src="__BASE_URI__images/equationeditor5.PNG" />to bring up an empty matrix. Next, replace each letter with your numeric answer, tabbing between cells.)</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;<span>Note: Please enter the values <em>exactly</em> - ie. as fractions or whole numbers, not as decimal numbers.</span></p><p>&nbsp;</p>@

qu.3.15.mode=Inline@
qu.3.15.name=Det of a 4x4 matrix@
qu.3.15.comment=<p>The determinant of $A is $det.</p>@
qu.3.15.editing=useHTML@
qu.3.15.solution=@
qu.3.15.algorithm=$m=maple("
randomize():
m1:=LinearAlgebra[RandomMatrix](4,generator=rand(-4..4),attributes=[nonsingular]):
m2:=LinearAlgebra[Determinant](m1):
m4:=MathML[ExportPresentation](m1):
m2,convert(m4,string)
");
$det=switch(0,$m);
$A=switch(1,$m);@
qu.3.15.uid=15cfab49-0060-4da0-ba99-e85061d0804e@
qu.3.15.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Determinants;
  Author=Katherine Dare;
  Difficulty=Hard;
@
qu.3.15.weighting=1@
qu.3.15.numbering=alpha@
qu.3.15.part.1.name=sro_id_1@
qu.3.15.part.1.answer.units=@
qu.3.15.part.1.numStyle=   @
qu.3.15.part.1.editing=useHTML@
qu.3.15.part.1.showUnits=false@
qu.3.15.part.1.question=(Unset)@
qu.3.15.part.1.mode=Numeric@
qu.3.15.part.1.grading=exact_value@
qu.3.15.part.1.negStyle=both@
qu.3.15.part.1.answer.num=$det@
qu.3.15.question=<p>What is the determinant of the following matrix?</p><p>$A</p><p>det(A)=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.4.topic=Cramer's Rule@

qu.4.1.mode=Inline@
qu.4.1.name=Cramer's Rule - steps@
qu.4.1.comment=<p>Cramer's rule states that: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>x</mi><mrow><mi>$k</mi></mrow></msub></mrow></mstyle></math>=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><msub><mi>A</mi><mrow><mi>$k</mi></mrow></msub><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mfenced open='|' close='|' separators=','><mrow><mi>A</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math></p> 
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>A</mi><mrow><mi>$k</mi></mrow></msub></mrow></mstyle></math>=$Ak</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><msub><mi>A</mi><mrow><mi>$k</mi></mrow></msub></mrow></mfenced></mrow></mstyle></math>=$DetAk</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>A</mi></mrow></mfenced></mrow></mstyle></math>=$DetA</p>
<p>Therefore<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>x</mi><mrow><mi>$k</mi></mrow></msub></mrow></mstyle></math>=$ans</p>@
qu.4.1.editing=useHTML@
qu.4.1.hint.1=Cramer's rule states that: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><msub><mi>A</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mfenced open='|' close='|' separators=','><mrow><mi>A</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>@
qu.4.1.solution=@
qu.4.1.algorithm=$k=range(1,3);
$v=maple("
randomize():
with(LinearAlgebra):
v1:=RandomMatrix(3,3,generator=(-3..3)):
while Determinant(v1) = 0 do v1:=RandomMatrix(3,3,generator=(-3..3)) end do:
v2:=RandomMatrix(3,1,generator=(-9..9)):
v3:=v1.v2:
v4:=MathML[ExportPresentation](v1):
v5:=MathML[ExportPresentation](v2):
v6:=MathML[ExportPresentation](v3):
v7:=v2[$k,1]:
v8:=Matrix(v1):
v8[1..3,$k..$k]:=v3:
v9:=MathML[ExportPresentation](v8):
v10:=Determinant(v8):
v11:=Determinant(v1):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(v5,string),convert(v6,string),v7,convert(v8,string),convert(v9,string),v10,v11
");
$A=switch(3,$v);
$ans=switch(6,$v);
$b=switch(5,$v);
$Akmath=switch(7,$v);
$Ak=switch(8,$v);
$DetAk=switch(9,$v);
$DetA=switch(10,$v);@
qu.4.1.uid=a2143873-5b2b-4841-b924-bc08c749ee70@
qu.4.1.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Solving A System With Cramer'S Rule;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Walks Students Through Steps;
@
qu.4.1.weighting=1,1,1,1@
qu.4.1.numbering=alpha@
qu.4.1.part.1.name=sro_id_1@
qu.4.1.part.1.answer.units=@
qu.4.1.part.1.numStyle=   @
qu.4.1.part.1.editing=useHTML@
qu.4.1.part.1.showUnits=false@
qu.4.1.part.1.question=(Unset)@
qu.4.1.part.1.mode=Numeric@
qu.4.1.part.1.grading=exact_value@
qu.4.1.part.1.negStyle=both@
qu.4.1.part.1.answer.num=$DetA@
qu.4.1.part.2.name=sro_id_2@
qu.4.1.part.2.maple_answer=printf("$Ak");@
qu.4.1.part.2.editing=useHTML@
qu.4.1.part.2.question=(Unset)@
qu.4.1.part.2.libname=@
qu.4.1.part.2.mode=Maple@
qu.4.1.part.2.allow2d=2@
qu.4.1.part.2.plot=@
qu.4.1.part.2.maple=ans:=$Akmath:
grade:=0:
for i from 1 to 3 do
for j from 1 to 3 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.11112:
end if;
end;
end;
grade;@
qu.4.1.part.2.type=maple@
qu.4.1.part.3.name=sro_id_3@
qu.4.1.part.3.answer.units=@
qu.4.1.part.3.numStyle=   @
qu.4.1.part.3.editing=useHTML@
qu.4.1.part.3.showUnits=false@
qu.4.1.part.3.question=(Unset)@
qu.4.1.part.3.mode=Numeric@
qu.4.1.part.3.grading=exact_value@
qu.4.1.part.3.negStyle=both@
qu.4.1.part.3.answer.num=$DetAk@
qu.4.1.part.4.name=sro_id_4@
qu.4.1.part.4.answer.units=@
qu.4.1.part.4.numStyle=   @
qu.4.1.part.4.editing=useHTML@
qu.4.1.part.4.showUnits=false@
qu.4.1.part.4.question=(Unset)@
qu.4.1.part.4.mode=Numeric@
qu.4.1.part.4.grading=exact_value@
qu.4.1.part.4.negStyle=both@
qu.4.1.part.4.answer.num=$ans@
qu.4.1.question=<p>Solve for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>x</mi><mrow><mi>$k</mi></mrow></msub></mrow></mstyle></math> using Cramer's Rule.</p><p>$A&nbsp; <font size="2"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>3</mn></mrow></msub></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math></font>&nbsp; =$b</p><p>Where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>=$A,</p><p>What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>A</mi></mrow></mfenced></mrow></mstyle></math>?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>A</mi></mrow></mfenced></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>A</mi><mrow><mi>$k</mi></mrow></msub></mrow></mstyle></math>?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>A</mi><mrow><mi>$k</mi></mrow></msub></mrow></mstyle></math>=<span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><msub><mi>A</mi><mrow><mi>$k</mi></mrow></msub></mrow></mfenced></mrow></mstyle></math>?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><msub><mi>A</mi><mrow><mi>$k</mi></mrow></msub></mrow></mfenced></mrow></mstyle></math>=<span>&nbsp;</span><3><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;Therefore, what is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>x</mi><mrow><mi>$k</mi></mrow></msub></mrow></mstyle></math>?</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>x</mi><mrow><mi>$k</mi></mrow></msub></mrow></mstyle></math>=<span>&nbsp;</span><4><span>&nbsp;</span></p>@

qu.4.2.mode=Inline@
qu.4.2.name=Solving a 3x3 system of equations - market clearing problem - Cramer's rule@
qu.4.2.comment=<p>Note: the method below solves for all prices. If you only need to find one, consider using Cramer's rule instead.</p>
<p>Equate each firm's demand to its supply and rearrange and the resulting system of equations can be written as the following matrix problem:</p>
<p>$Amatrix<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><msub><mi>P</mi><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>P</mi><mrow><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>P</mi><mrow><mn>3</mn></mrow></msub></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$constants</p>
<p>The inverse of $Amatrix is:</p>
<p>$inv</p>
<p>&nbsp;</p>
<p>Pre-multiplying both sides of the equation by the inverse gives:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><msub><mi>P</mi><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>P</mi><mrow><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>P</mi><mrow><mn>3</mn></mrow></msub></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$inv<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow></mstyle></math>$constants=$prices</p>@
qu.4.2.editing=useHTML@
qu.4.2.hint.1=Equate each firm's supply and demand.@
qu.4.2.hint.2=Rearrange so that all the variables are on the left hand side and all the constants are on the right hand side.@
qu.4.2.hint.3=Translate system into matrix form.@
qu.4.2.hint.4=Use Cramer's rule to solve.@
qu.4.2.solution=@
qu.4.2.algorithm=$k=range(1,3);
$a=range(1,2);
$b=range(1,2);
$c=range(1,2);
$v=maple("
randomize():
v1:=LinearAlgebra[RandomMatrix](3,generator=rand(3..6),attributes=[nonsingular]):
v1[1..1,1..1]:=(-1)*v1[1,1]:
v1[2..2,2..2]:=(-1)*v1[2,2]:
v1[3..3,3..3]:=(-1)*v1[3,3]:
while LinearAlgebra[Determinant](v1) = 0 do v1 := LinearAlgebra[RandomMatrix](3, generator = rand(1 .. 5), attributes = [nonsingular]): 
v1[1..1,1..1]:=(-1)*v1[1,1]:
v1[2..2,2..2]:=(-1)*v1[2,2]:
v1[3..3,3..3]:=(-1)*v1[3,3]:
end do:
v2:=LinearAlgebra[RandomMatrix](3,1,generator=rand(5..15)):
v3:=v1.v2:
v4:=MathML[ExportPresentation](v1):
v6:=MathML[ExportPresentation](v1.v2):
q1d:=MathML[ExportPresentation](-(v3[1,1])+(v1[1,1]+$a)*P1+v1[1,2]*P2+v1[1,3]*P3):
q2d:=MathML[ExportPresentation](-(v3[2,1])+v1[2,1]*P1+(v1[2,2]+$b)*P2+v1[2,3]*P3):
q3d:=MathML[ExportPresentation](-(v3[3,1])+v1[3,1]*P1+v1[3,2]*P2+(v1[3,3]+$c)*P3):
q1s:=MathML[ExportPresentation]($a*P1):
q2s:=MathML[ExportPresentation]($b*P2):
q3s:=MathML[ExportPresentation]($c*P3):
P1:=v2[1,1]:
P2:=v2[2,1]:
P3:=v2[3,1];
inv:=LinearAlgebra[MatrixInverse](v1):
invpretty:=MathML[ExportPresentation](inv):
prices:=MathML[ExportPresentation](v2):
constants:=MathML[ExportPresentation](v3):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(q1d,string),convert(q2d,string),convert(q3d,string),convert(q1s,string),convert(q2s,string),convert(q3s,string),P1,P2,P3,invpretty,prices,constants
");
$q1d=switch(4,$v);
$q2d=switch(5,$v);
$q3d=switch(6,$v);
$q1s=switch(7,$v);
$q2s=switch(8,$v);
$q3s=switch(9,$v);
$P1=switch(10,$v);
$P2=switch(11,$v);
$P3=switch(12,$v);
$inv=switch(13,$v);
$prices=switch(14,$v);
$Amatrix=switch(3,$v);
$constants=switch(15,$v);
$ans=switch($k-1,$P1,$P2,$P3);@
qu.4.2.uid=43118d26-8db8-4650-8e17-7c59a90c8083@
qu.4.2.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Solving A System Of Equations;
  Author=Katherine Dare;
  Difficulty=Hard;
@
qu.4.2.weighting=1@
qu.4.2.numbering=alpha@
qu.4.2.part.1.name=sro_id_1@
qu.4.2.part.1.answer.units=@
qu.4.2.part.1.numStyle=   @
qu.4.2.part.1.editing=useHTML@
qu.4.2.part.1.showUnits=false@
qu.4.2.part.1.question=(Unset)@
qu.4.2.part.1.mode=Numeric@
qu.4.2.part.1.grading=exact_value@
qu.4.2.part.1.negStyle=both@
qu.4.2.part.1.answer.num=$ans@
qu.4.2.question=<p>Three firms sell related products in a market. Their supply functions are:</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>q</mi><mrow><mn>1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$q1s</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>q</mi><mrow><mn>2</mn></mrow><mrow><mi>s</mi></mrow></msubsup></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$q2s</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>q</mi><mrow><mn>3</mn></mrow><mrow><mi>s</mi></mrow></msubsup></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$q3s</p><p>&nbsp;</p><p>The firms face the following demands:</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>q</mi><mrow><mn>1</mn></mrow><mrow><mi>d</mi></mrow></msubsup></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$q1d</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>q</mi><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$q2d</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>q</mi><mrow><mn>3</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$q3d</p><p>&nbsp;</p><p>&nbsp;</p><p><span>Given the above supply and demand functions, what price must firm $k charge to equate supply and demand? Note that you do not need to solve for all equilibrium prices. <br /></span></p><p>&nbsp;</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>P</mi><mrow><mi mathvariant='normal'>$k</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.4.3.mode=Inline@
qu.4.3.name=Cramer's Rule@
qu.4.3.comment=<p>Cramer's rule states that: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>x</mi><mrow><mi>$k</mi></mrow></msub></mrow></mstyle></math>=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><msub><mi>A</mi><mrow><mi>$k</mi></mrow></msub><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mfenced open='|' close='|' separators=','><mrow><mi>A</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math></p> 
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>A</mi><mrow><mi>$k</mi></mrow></msub></mrow></mstyle></math>=$Ak</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><msub><mi>A</mi><mrow><mi>$k</mi></mrow></msub></mrow></mfenced></mrow></mstyle></math>=$DetAk</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>=$A</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>A</mi></mrow></mfenced></mrow></mstyle></math>=$DetA</p>
<p>Therefore<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>x</mi><mrow><mi>$k</mi></mrow></msub></mrow></mstyle></math>=$ans</p>@
qu.4.3.editing=useHTML@
qu.4.3.hint.1=Cramer's rule states that: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><msub><mi>A</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mfenced open='|' close='|' separators=','><mrow><mi>A</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>@
qu.4.3.solution=@
qu.4.3.algorithm=$k=range(1,3);
$v=maple("
randomize():
with(LinearAlgebra):
v1:=RandomMatrix(3,3,generator=(-3..3)):
while LinearAlgebra[Determinant](v1) = 0 do v1:=RandomMatrix(3,3,generator=(-3..3)) end do:
v2:=RandomMatrix(3,1,generator=(-9..9)):
v3:=v1.v2:
v4:=MathML[ExportPresentation](v1):
v5:=MathML[ExportPresentation](v2):
v6:=MathML[ExportPresentation](v3):
v7:=v2[$k,1]:
v8:=Matrix(v1):
v8[1..3,$k..$k]:=v3:
v9:=MathML[ExportPresentation](v8):
v10:=Determinant(v8):
v11:=Determinant(v1):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(v5,string),convert(v6,string),v7,convert(v8,string),convert(v9,string),v10,v11
");
$A=switch(3,$v);
$ans=switch(6,$v);
$b=switch(5,$v);
$Ak=switch(8,$v);
$DetAk=switch(9,$v);
$DetA=switch(10,$v);@
qu.4.3.uid=0b014d7a-4799-435b-9a40-918ea72d5c4b@
qu.4.3.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Solving A System With Cramer'S Rule;
  Author=Katherine Dare;
  Difficulty=Medium;
@
qu.4.3.weighting=1@
qu.4.3.numbering=alpha@
qu.4.3.part.1.name=sro_id_1@
qu.4.3.part.1.answer.units=@
qu.4.3.part.1.numStyle=   @
qu.4.3.part.1.editing=useHTML@
qu.4.3.part.1.showUnits=false@
qu.4.3.part.1.question=(Unset)@
qu.4.3.part.1.mode=Numeric@
qu.4.3.part.1.grading=exact_value@
qu.4.3.part.1.negStyle=both@
qu.4.3.part.1.answer.num=$ans@
qu.4.3.question=<p>Solve for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>x</mi><mrow><mi>$k</mi></mrow></msub></mrow></mstyle></math> using Cramer's Rule.</p><p>$A&nbsp; <font size="2"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>3</mn></mrow></msub></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math></font>&nbsp; =$b</p><p>&nbsp;</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>x</mi><mrow><mi>$k</mi></mrow></msub></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.5.topic=Applications@

qu.5.1.mode=Inline@
qu.5.1.name=Input Requirements Matrix - 2x2 multiplication@
qu.5.1.comment=<p>$A*$y=$z</p>
<p>Therefore the bakery needs $ans1 units of flour and $ans2 units of sugar.</p>@
qu.5.1.editing=useHTML@
qu.5.1.solution=@
qu.5.1.algorithm=$v=maple("
randomize():
v1:=LinearAlgebra[RandomMatrix](2,generator=(1..4)):
v2:=LinearAlgebra[RandomMatrix](2,1,generator=(3..8)):
v3:=v1.v2:
v4:=MathML[ExportPresentation](v1):
v5:=MathML[ExportPresentation](v2):
v6:=MathML[ExportPresentation](v3):
v7:=v1[1,1]:
v8:=v1[2,1]:
v9:=v1[1,2]:
v10:=v1[2,2]:
v11:=v2[1,1]:
v12:=v2[2,1]:
v13:=v3[1,1]:
v14:=v3[2,1]:
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(v5,string),convert(v6,string),v7,v8,v9,v10,v11,v12,v13,v14
");
$A=switch(3,$v);
$y=switch(4,$v);
$z=switch(5,$v);
$a11=switch(6,$v);
$a21=switch(7,$v);
$a12=switch(8,$v);
$a22=switch(9,$v);
$output1=switch(10,$v);
$output2=switch(11,$v);
$ans1=switch(12,$v);
$ans2=switch(13,$v);@
qu.5.1.uid=0031cc63-3540-440b-a980-ba6f7c6411b1@
qu.5.1.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Matrix Multiplication - Input Requirements Matrix;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Students Use Equation Editor;
@
qu.5.1.weighting=1,1@
qu.5.1.numbering=alpha@
qu.5.1.part.1.name=sro_id_1@
qu.5.1.part.1.answer.units=@
qu.5.1.part.1.numStyle=   @
qu.5.1.part.1.editing=useHTML@
qu.5.1.part.1.showUnits=false@
qu.5.1.part.1.question=(Unset)@
qu.5.1.part.1.mode=Numeric@
qu.5.1.part.1.grading=exact_value@
qu.5.1.part.1.negStyle=both@
qu.5.1.part.1.answer.num=$ans1@
qu.5.1.part.2.name=sro_id_2@
qu.5.1.part.2.answer.units=@
qu.5.1.part.2.numStyle=   @
qu.5.1.part.2.editing=useHTML@
qu.5.1.part.2.showUnits=false@
qu.5.1.part.2.question=(Unset)@
qu.5.1.part.2.mode=Numeric@
qu.5.1.part.2.grading=exact_value@
qu.5.1.part.2.negStyle=both@
qu.5.1.part.2.answer.num=$ans2@
qu.5.1.question=<p>Julie is trying to make some bread and cake and uses an input requirement matrix to help her buy her ingredients. An input requirement matrix represents how many inputs are needed to make one unit of different outputs. For example, the following matrix:</p><p>&nbsp;</p><p>$A</p><p>&nbsp;</p><p>says that it takes Julie $a11 cups of flour and $a21 cups of sugar to make one loaf of bread. Similarly, it takes her $a12 cups of flour and $a22 cups of sugar to make one cake.&nbsp; (Julie is not a very good baker and therefore doesn't know the proper ratios.)</p><p>&nbsp;</p><p>If Julie wants to make <strong>$output1 loaves of bread</strong> and <strong>$output2 cakes</strong>, how much of each input does she need?</p><p>&nbsp;</p><p>(Do not enter units.)</p><p>Flour=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>Sugar=<span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.5.2.mode=Inline@
qu.5.2.name=Migration - 3x3 transition matrix@
qu.5.2.comment=<p>&nbsp;$transition*$pop=$ansvector</p>
<p>Therefore, $ans1 people end up in Guelph,$ans2 people end up in Waterloo and $ans3 people end up in Toronto.</p>@
qu.5.2.editing=useHTML@
qu.5.2.solution=@
qu.5.2.algorithm=$a11=range(1,6)*0.1;
$a22=range(1,6)*0.1;
$a33=range(1,6)*0.1;
$a21=range(1,4)*0.1;
$a12=range(1,4)*0.1;
$a13=range(1,4)*0.1;
$a31=1-$a11-$a21;
$a32=1-$a22-$a12;
$a23=1-$a33-$a13;
$a11percent=$a11*100;
$a21percent=$a21*100;
$a31percent=$a31*100;
$a12percent=$a12*100;
$a22percent=$a21*100;
$a32percent=$a32*100;
$v=maple("
randomize():
v1:=Matrix([[$a11,$a12,$a13],[$a21,$a22,$a23],[$a31,$a32,$a33]]):
v2:=LinearAlgebra[RandomMatrix](3,1,generator=(5..20)):
v3:=v1.v2:
v4:=MathML[ExportPresentation](v1):
v5:=MathML[ExportPresentation](v2):
v6:=MathML[ExportPresentation](v3):
v7:=v3[1,1]:
v8:=v3[2,1]:
v9:=v3[3,1]:
v10:=v2[1,1]:
v11:=v2[2,1]:
v12:=v2[3,1]:
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(v5,string),convert(v6,string),v7,v8,v9,v10,v11,v12
");
$transition=switch(3,$v);
$pop=switch(4,$v);
$ansvector=switch(5,$v);
$ans1=switch(6,$v);
$ans2=switch(7,$v);
$ans3=switch(8,$v);
$pop1=switch(9,$v);
$pop2=switch(10,$v);
$pop3=switch(11,$v);@
qu.5.2.uid=c6b4f6a1-e7fe-44ba-aa75-851acf470f0b@
qu.5.2.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Matrix Multiplication, Migration;
  Author=Katherine Dare;
  Difficulty=Medium;
@
qu.5.2.weighting=1,1,1@
qu.5.2.numbering=alpha@
qu.5.2.part.1.name=sro_id_1@
qu.5.2.part.1.answer.units=@
qu.5.2.part.1.numStyle=   @
qu.5.2.part.1.editing=useHTML@
qu.5.2.part.1.showUnits=false@
qu.5.2.part.1.question=(Unset)@
qu.5.2.part.1.mode=Numeric@
qu.5.2.part.1.grading=exact_value@
qu.5.2.part.1.negStyle=both@
qu.5.2.part.1.answer.num=$ans1@
qu.5.2.part.2.name=sro_id_2@
qu.5.2.part.2.answer.units=@
qu.5.2.part.2.numStyle=   @
qu.5.2.part.2.editing=useHTML@
qu.5.2.part.2.showUnits=false@
qu.5.2.part.2.question=(Unset)@
qu.5.2.part.2.mode=Numeric@
qu.5.2.part.2.grading=exact_value@
qu.5.2.part.2.negStyle=both@
qu.5.2.part.2.answer.num=$ans2@
qu.5.2.part.3.name=sro_id_3@
qu.5.2.part.3.answer.units=@
qu.5.2.part.3.numStyle=   @
qu.5.2.part.3.editing=useHTML@
qu.5.2.part.3.showUnits=false@
qu.5.2.part.3.question=(Unset)@
qu.5.2.part.3.mode=Numeric@
qu.5.2.part.3.grading=exact_value@
qu.5.2.part.3.negStyle=both@
qu.5.2.part.3.answer.num=$ans3@
qu.5.2.question=<p>The flow of people from one place to another can be described by a transition matrix like the following:</p><p>&nbsp;</p><p>$transition</p><p>&nbsp;</p><p>This matrix shows that $a11percent percent of the people currently in the first region (call it Guelph) stay there, while $a21percent percent of them move to the second region (call it Waterloo) and $a31percent percent move to the third region (call it Toronto). Similarly, $a22percent percent of people in Waterloo stay, $a12percent percent move to Guelph and $a32percent percent move to Toronto.</p><p>&nbsp;</p><p>If the number of thousand people in <strong>Guelph before</strong> <strong>the move is</strong> <strong>$pop1</strong>, the number of thousand people in <strong>Waterloo before the move is</strong> <strong>$pop2</strong>, and the number of people in <strong>Toronto before the movie is $pop3</strong>, how many thousand people are in each region after the move?</p><p>&nbsp;</p><p>(Answers may be decimal numbers. Do not round.)</p><p>Guelph=<span>&nbsp;</span><1><span>&nbsp; thousand people<br /></span></p><p>Waterloo=<span>&nbsp;</span><2><span>&nbsp; thousand people<br /></span></p><p><span>Toronto=<span>&nbsp;</span><3><span>&nbsp; thousand people<br /></span></span></p><p>&nbsp;</p>@

qu.5.3.mode=Inline@
qu.5.3.name=Migration - 2x2 transition matrix@
qu.5.3.comment=<p>&nbsp;$transition*$pop=$ansvector</p>
<p>Therefore, $ans1 people end up in Guelph and $ans2 people end up in Waterloo.</p>@
qu.5.3.editing=useHTML@
qu.5.3.solution=@
qu.5.3.algorithm=$a=range(1,9)*0.1;
$b=range(1,9)*0.1;
$c=$a*100;
$d=(1-$a)*100;
$e=$b*100;
$f=(1-$b)*100;
$v=maple("
randomize():
v1:=Matrix([[$a,$b],[1-$a,1-$b]]):
v2:=LinearAlgebra[RandomMatrix](2,1,generator=(5..20)):
v3:=v1.v2:
v4:=MathML[ExportPresentation](v1):
v5:=MathML[ExportPresentation](v2):
v6:=MathML[ExportPresentation](v3):
v7:=v3[1,1]:
v8:=v3[2,1]:
v9:=v2[1,1]:
v10:=v2[2,1]:
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(v5,string),convert(v6,string),v7,v8,v9,v10
");
$transition=switch(3,$v);
$pop=switch(4,$v);
$ansvector=switch(5,$v);
$ans1=switch(6,$v);
$ans2=switch(7,$v);
$pop1=switch(8,$v);
$pop2=switch(9,$v);@
qu.5.3.uid=94a771df-0106-4080-93bb-475e696d51fa@
qu.5.3.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Matrix Multiplication, Migration;
  Author=Katherine Dare;
  Difficulty=Easy;
@
qu.5.3.weighting=1,1@
qu.5.3.numbering=alpha@
qu.5.3.part.1.name=sro_id_1@
qu.5.3.part.1.answer.units=@
qu.5.3.part.1.numStyle=   @
qu.5.3.part.1.editing=useHTML@
qu.5.3.part.1.showUnits=false@
qu.5.3.part.1.question=(Unset)@
qu.5.3.part.1.mode=Numeric@
qu.5.3.part.1.grading=exact_value@
qu.5.3.part.1.negStyle=both@
qu.5.3.part.1.answer.num=$ans1@
qu.5.3.part.2.name=sro_id_2@
qu.5.3.part.2.answer.units=@
qu.5.3.part.2.numStyle=   @
qu.5.3.part.2.editing=useHTML@
qu.5.3.part.2.showUnits=false@
qu.5.3.part.2.question=(Unset)@
qu.5.3.part.2.mode=Numeric@
qu.5.3.part.2.grading=exact_value@
qu.5.3.part.2.negStyle=both@
qu.5.3.part.2.answer.num=$ans2@
qu.5.3.question=<p>The flow of people from one place to another can be described by a transition matrix like the following:</p><p>&nbsp;</p><p>$transition</p><p>&nbsp;</p><p>This matrix shows that $c percent of the people currently in the first region (call it Guelph) stay there, while $d percent of them move to the second region (call it Waterloo). Similarly, $f percent of people in Waterloo stay, and $e percent move to Guelph.</p><p>&nbsp;</p><p>If the number of thousand people in <strong>Guelph before</strong> <strong>the move is</strong> <strong>$pop1</strong>, and the number of thousand people in <strong>Waterloo before the move is</strong> <strong>$pop2</strong>, how many people are in each region after the move?</p><p>&nbsp;</p><p>(Answers may be decimal numbers. Do not round.)</p><p>Guelph=<span>&nbsp;</span><1><span>&nbsp; thousand people<br /></span></p><p>Waterloo=<span>&nbsp;</span><2><span>&nbsp; thousand people<br /></span></p><p>&nbsp;</p>@

qu.5.4.mode=Inline@
qu.5.4.name=Input Requirements Matrix - 2x2 inverse@
qu.5.4.comment=<p>The inverse of the input requirements matrix is:</p>
<p>$invA</p>
<p>and</p>
<p>$invA*$z=$y</p>
<p>Therefore the bakery can make $ans1 loaves of bread and $ans2 cakes.</p>@
qu.5.4.editing=useHTML@
qu.5.4.solution=@
qu.5.4.algorithm=$v=maple("
randomize():
v1:=LinearAlgebra[RandomMatrix](2,generator=(1..4)):
while LinearAlgebra[Determinant](v1) = 0 do v1 := LinearAlgebra[RandomMatrix](2, generator = (1..4), attributes = [nonsingular]) end do:
v2:=LinearAlgebra[RandomMatrix](2,1,generator=(3..8)):
v3:=v1.v2:
v4:=MathML[ExportPresentation](v1):
v5:=MathML[ExportPresentation](v2):
v6:=MathML[ExportPresentation](v3):
v7:=v1[1,1]:
v8:=v1[2,1]:
v9:=v1[1,2]:
v10:=v1[2,2]:
v11:=v2[1,1]:
v12:=v2[2,1]:
v13:=v3[1,1]:
v14:=v3[2,1]:
v15:=MathML[ExportPresentation](LinearAlgebra[MatrixInverse](v1)):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(v5,string),convert(v6,string),v7,v8,v9,v10,v11,v12,v13,v14,convert(v15,string)
");
$A=switch(3,$v);
$y=switch(4,$v);
$z=switch(5,$v);
$a11=switch(6,$v);
$a21=switch(7,$v);
$a12=switch(8,$v);
$a22=switch(9,$v);
$ans1=switch(10,$v);
$ans2=switch(11,$v);
$input1=switch(12,$v);
$input2=switch(13,$v);
$invA=switch(14,$v);@
qu.5.4.uid=bafda39a-a23c-47b0-bdb2-5f4cdb710188@
qu.5.4.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Matrix Multiplication - Input Requirements Matrix;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Students Use Equation Editor;
@
qu.5.4.weighting=1,1@
qu.5.4.numbering=alpha@
qu.5.4.part.1.name=sro_id_1@
qu.5.4.part.1.answer.units=@
qu.5.4.part.1.numStyle=   @
qu.5.4.part.1.editing=useHTML@
qu.5.4.part.1.showUnits=false@
qu.5.4.part.1.question=(Unset)@
qu.5.4.part.1.mode=Numeric@
qu.5.4.part.1.grading=exact_value@
qu.5.4.part.1.negStyle=both@
qu.5.4.part.1.answer.num=$ans1@
qu.5.4.part.2.name=sro_id_2@
qu.5.4.part.2.answer.units=@
qu.5.4.part.2.numStyle=   @
qu.5.4.part.2.editing=useHTML@
qu.5.4.part.2.showUnits=false@
qu.5.4.part.2.question=(Unset)@
qu.5.4.part.2.mode=Numeric@
qu.5.4.part.2.grading=exact_value@
qu.5.4.part.2.negStyle=both@
qu.5.4.part.2.answer.num=$ans2@
qu.5.4.question=<p>Julie is trying to make some bread and cake and uses an input  requirement matrix to help her figure out how much she can make. An input requirement  matrix represents how many inputs are needed to make one unit of  different outputs. For example, the following matrix:</p><p>&nbsp;</p><p>$A</p><p>&nbsp;</p><p>says  that it takes Julie $a11 cups of flour and $a21 cups of sugar to make  one loaf of bread. Similarly, it takes her $a12 cups of flour and $a22  cups of sugar to make one cake.&nbsp; (Julie is not a very good baker and therefore doesn't know the proper ratios.)</p><p>&nbsp;</p><p>&nbsp;</p><p>If Julie has <strong>$input1 units of flour</strong> and <strong>$input2 units of sugar</strong>, how much bread and cake can she make?</p><p>&nbsp;</p><p>(Do not enter units.)</p><p>Cake=<span>&nbsp;</span><1><span>&nbsp;</span></p><p>Bread=<span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.6.topic=Gauss Jordan@

qu.6.1.mode=Inline@
qu.6.1.name=Gauss Jordan@
qu.6.1.comment=<p>Note:&nbsp; There is no <em>unique</em> solution to this problem.</p>
<p>For this question we need to do row operations until we have zeros and ones in the required locations. This means that you can interchange rows or add any multiple of a row to another row or multiply a row by a constant.</p>
<p>If you were to solve the system completely, the solution would be $xpretty.</p>@
qu.6.1.editing=useHTML@
qu.6.1.solution=@
qu.6.1.algorithm=$v=maple("
randomize():
v1:=LinearAlgebra[RandomMatrix](3,generator=rand(-3..3),attributes=[nonsingular]):
while LinearAlgebra[Determinant](v1) = 0 do v1 := LinearAlgebra[RandomMatrix](3, generator = rand(-3 .. 3), attributes = [nonsingular]) end do:
v2:=LinearAlgebra[Determinant](v1):
v3:=MathML[ExportPresentation](LinearAlgebra[MatrixInverse](v1)):
v4:=MathML[ExportPresentation](v1):
v5:=LinearAlgebra[MatrixInverse](v1):
v6:=LinearAlgebra[RandomMatrix](3,1,generator=rand(2..5)):
v7:=v1.v6:
v8:=MathML[ExportPresentation](v7):
v9:=MathML[ExportPresentation](v6):
v10:=Matrix([[1,0,0,0],[0,1,0,0],[0,0,1,0]]):
v10[1..3,4..4]:=v6:
v11:=MathML[ExportPresentation](v10):
v12:=Matrix([Matrix(3,3,shape=diagonal,fill=1),v6]):
v2,v3,v4,convert(v5,string),convert(v6,string),convert(v7,string),v8,v9,v11,convert(v12,string)
");
$inv=switch(1,$v);
$A=switch(2,$v);
$xmath=switch(4,$v);
$bmath=switch(5,$v);
$b=switch(6,$v);
$xpretty=switch(8,$v);
$ans=switch(10,$v);@
qu.6.1.uid=5fd861c1-7d24-4928-acf7-9ec0d84d7636@
qu.6.1.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Gauss Jordan Elimination;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Students Use Equation Editor;
@
qu.6.1.weighting=1@
qu.6.1.numbering=alpha@
qu.6.1.part.1.name=sro_id_1@
qu.6.1.part.1.maple_answer=$ans@
qu.6.1.part.1.editing=useHTML@
qu.6.1.part.1.question=(Unset)@
qu.6.1.part.1.libname=@
qu.6.1.part.1.mode=Maple@
qu.6.1.part.1.allow2d=2@
qu.6.1.part.1.plot=@
qu.6.1.part.1.maple=ans:=$xmath:
grade:=0:
if $RESPONSE[1,1]=1
then
if $RESPONSE[2,2]=1
then
if $RESPONSE[2,1]=0
then
for i from 1 to 3 do
temp1:=$RESPONSE[i,1]*ans[1,1]:
temp2:=$RESPONSE[i,2]*ans[2,1]:
temp3:=$RESPONSE[i,3]*ans[3,1]:
temp4:=temp1+temp2+temp3:
if temp4=$RESPONSE[i,4]
then grade:=grade+0.33334:
end if;
end;
end if;
end if;
end if;


grade;@
qu.6.1.part.1.type=maple@
qu.6.1.question=<p>$A<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>3</mn></mrow></msub></mrow></mtd></mtr></mtable></mfenced></mrow></mrow></mstyle></math>=$b</p><p>&nbsp;</p><p>Use Gauss Jordan to solve the above system of equations. You only need to transform your matrix to the point where it looks like this:</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>1</mn></mrow></mtd><mtd><mrow><mo lspace='0.0em' rspace='0.0em'>&num;</mo></mrow></mtd><mtd><mrow><mo lspace='0.0em' rspace='0.0em'>&num;</mo></mrow></mtd><mtd><mrow><mo lspace='0.0em' rspace='0.0em'>&num;</mo></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>1</mn></mrow></mtd><mtd><mrow><mo lspace='0.0em' rspace='0.0em'>&num;</mo></mrow></mtd><mtd><mrow><mo lspace='0.0em' rspace='0.0em'>&num;</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo lspace='0.0em' rspace='0.0em'>&num;</mo></mrow></mtd><mtd><mrow><mo lspace='0.0em' rspace='0.0em'>&num;</mo></mrow></mtd><mtd><mrow><mo lspace='0.0em' rspace='0.0em'>&num;</mo></mrow></mtd><mtd><mrow><mo lspace='0.0em' rspace='0.0em'>&num;</mo></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math></p><p>(# means the values that result from row operations)</p><p><span>&nbsp;</span><1><span>&nbsp; Note: Your answer should be a 3x4 matrix. <br /></span></p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p>@

qu.7.topic=Matrix Types@

qu.7.1.mode=Multiple Selection@
qu.7.1.name=Matrix types - diagonal indefinite@
qu.7.1.comment=@
qu.7.1.editing=useHTML@
qu.7.1.solution=@
qu.7.1.algorithm=$u=range(1,9,1);
$v=range(1,9,1);
$w=range(1,9,1);
$d=range(0,1,1);
$e=range(0,1,1);
$f=(((2*($d))-1)*((2*($e))-1)+1)/2;
$g=range(0,1,1);
$h=(-1)*($d)+1;
$a=(-1)^($d)*($u);
$b=(-1)^($e)*($v);
$c=(-1)^(switch(($f),($g),($h)))*($w);
$A=maple("
A1:=Matrix([[$a,0,0],[0,$b,0],[0,0,$c]]):
A2:=MathML[ExportPresentation](A1):
A2
");
$Q=switch(0,$A);@
qu.7.1.uid=0f199c8b-b3bd-48e2-a4ed-1132ca573f80@
qu.7.1.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Types;
  Sub-Topic=Diagonal Indefinite;
  Author=Asha Sadanand;
  Difficulty=Easy;
  Feature=Multiple Selection;
@
qu.7.1.question=<p>Consider the following matrix.</p>
<p>&nbsp;$Q</p>
<p>Choose all the terms that apply to this matrix.</p>@
qu.7.1.answer=1, 3, 5, 6, 9, 10@
qu.7.1.choice.1=invertible@
qu.7.1.choice.2=identity@
qu.7.1.choice.3=full rank@
qu.7.1.choice.4=idempotent@
qu.7.1.choice.5=symmetric@
qu.7.1.choice.6=triangular@
qu.7.1.choice.7=zero matrix@
qu.7.1.choice.8=negative definite@
qu.7.1.choice.9=indefinite@
qu.7.1.choice.10=diagonal matrix@
qu.7.1.choice.11=positive definite@
qu.7.1.fixed=@

qu.7.2.mode=Multiple Selection@
qu.7.2.name=Matrix types - non-diagonal indefinite@
qu.7.2.comment=@
qu.7.2.editing=useHTML@
qu.7.2.solution=@
qu.7.2.algorithm=$u=range(1,9,1);
$v=range(1,9,1);
$w=range(1,9,1);
$d=range(0,1,1);
$e=range(0,1,1);
$f=(((2*($d))-1)*((2*($e))-1)+1)/2;
$g=range(0,1,1);
$h=(-1)*($d)+1;
$a=(-1)^($d)*($u);
$b=(-1)^($e)*($v);
$c=(-1)^(switch(($f),($g),($h)))*($w);
$A=maple("
A0:=Matrix([[$a,0,0],[0,$b,0],[0,0,$c]]):
m1:=LinearAlgebra[RandomMatrix](3,generator=rand(1..3),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m1) = 0 do m1 := LinearAlgebra[RandomMatrix](3, generator = rand(1 .. 3), attributes = [nonsingular]) end do:
m2:=LinearAlgebra[Determinant](m1):
m5:=LinearAlgebra[MatrixInverse](m1):
A1:=m2*(m5.A0.m1):
A2:=MathML[ExportPresentation](A1):
A2
");
$Q=switch(0,$A);@
qu.7.2.uid=189b42d6-6aad-4968-be23-59f745854351@
qu.7.2.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Types;
  Sub-Topic=Diagonal Positive Definite;
  Author=Asha Sadanand;
  Difficulty=Easy;
  Feature=Multiple Selection;
@
qu.7.2.question=<p>Consider the following matrix.</p>
<p>&nbsp;$Q</p>
<p>Choose all the terms that apply to this matrix.</p>@
qu.7.2.answer=1, 3, 9@
qu.7.2.choice.1=invertible@
qu.7.2.choice.2=identity@
qu.7.2.choice.3=full rank@
qu.7.2.choice.4=idempotent@
qu.7.2.choice.5=symmetric@
qu.7.2.choice.6=triangular@
qu.7.2.choice.7=zero matrix@
qu.7.2.choice.8=negative definite@
qu.7.2.choice.9=indefinite@
qu.7.2.choice.10=diagonal matrix@
qu.7.2.choice.11=positive definite@
qu.7.2.fixed=@

qu.7.3.mode=Multiple Selection@
qu.7.3.name=Matrix types - diagonal negative definite@
qu.7.3.comment=@
qu.7.3.editing=useHTML@
qu.7.3.solution=@
qu.7.3.algorithm=$a=-1*range(1,9,1);
$b=-1*range(1,9,1);
$c=-1*range(1,9,1);
$A=maple("
A1:=Matrix([[$a,0,0],[0,$b,0],[0,0,$c]]):
A2:=MathML[ExportPresentation](A1):
A2
");
$Q=switch(0,$A);@
qu.7.3.uid=749c73e0-7399-41d1-9e18-7d956181bd82@
qu.7.3.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Types;
  Sub-Topic=Diagonal Negative Definite;
  Author=Asha Sadanand;
  Difficulty=Easy;
  Feature=Multiple Selection;
@
qu.7.3.question=<p>Consider the following matrix.</p>
<p>&nbsp;$Q</p>
<p>Choose all the terms that apply to this matrix.</p>@
qu.7.3.answer=1, 3, 5, 6, 8, 10@
qu.7.3.choice.1=invertible@
qu.7.3.choice.2=identity@
qu.7.3.choice.3=full rank@
qu.7.3.choice.4=idempotent@
qu.7.3.choice.5=symmetric@
qu.7.3.choice.6=triangular@
qu.7.3.choice.7=zero matrix@
qu.7.3.choice.8=negative definite@
qu.7.3.choice.9=indefinite@
qu.7.3.choice.10=diagonal matrix@
qu.7.3.choice.11=positive definite@
qu.7.3.fixed=@

qu.7.4.mode=Multiple Selection@
qu.7.4.name=Matrix types - non-diagonal positive definite@
qu.7.4.comment=@
qu.7.4.editing=useHTML@
qu.7.4.solution=@
qu.7.4.algorithm=$a=range(1,5,1);
$b=range(1,5,1);
$c=range(1,5,1);
$A=maple("
m1:=LinearAlgebra[RandomMatrix](3,generator=rand(1..3),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m1) = 0 do m1 := LinearAlgebra[RandomMatrix](3, generator = rand(1 .. 3), attributes = [nonsingular]) end do:
m2:=LinearAlgebra[Determinant](m1):
m5:=LinearAlgebra[MatrixInverse](m1):
A0:=Matrix([[$a,0,0],[0,$b,0],[0,0,$c]]):
A1:=m2*(m5.A0.m1):
A2:=MathML[ExportPresentation](A1):
A2
");
$Q=switch(0,$A);@
qu.7.4.uid=04996af4-b2db-4851-a374-9e780447e826@
qu.7.4.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Types;
  Sub-Topic=Non-Diagonal Positive Definite;
  Author=Asha Sadanand;
  Difficulty=Medium;
  Feature=Multiple Selection;
@
qu.7.4.question=<p>Consider the following matrix.</p>
<p>&nbsp;$Q</p>
<p>Choose all the terms that apply to this matrix.</p>@
qu.7.4.answer=1, 3, 11@
qu.7.4.choice.1=invertible@
qu.7.4.choice.2=identity@
qu.7.4.choice.3=full rank@
qu.7.4.choice.4=idempotent@
qu.7.4.choice.5=symmetric@
qu.7.4.choice.6=triangular@
qu.7.4.choice.7=zero matrix@
qu.7.4.choice.8=negative definite@
qu.7.4.choice.9=indefinite@
qu.7.4.choice.10=diagonal matrix@
qu.7.4.choice.11=positive definite@
qu.7.4.fixed=@

qu.7.5.mode=Multiple Selection@
qu.7.5.name=Matrix types - diagonal positive definite@
qu.7.5.comment=@
qu.7.5.editing=useHTML@
qu.7.5.solution=@
qu.7.5.algorithm=$a=range(1,9,1);
$b=range(1,9,1);
$c=range(1,9,1);
$A=maple("
A1:=Matrix([[$a,0,0],[0,$b,0],[0,0,$c]]):
A2:=MathML[ExportPresentation](A1):
A2
");
$Q=switch(0,$A);@
qu.7.5.uid=0f6d16c1-a4d1-45d3-8fb4-a7773156ff90@
qu.7.5.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Types;
  Sub-Topic=Diagonal Positive Definite;
  Author=Asha Sadanand;
  Difficulty=Easy;
  Feature=Multiple Selection;
@
qu.7.5.question=<p>Consider the following matrix.</p>
<p>&nbsp;$Q</p>
<p>Choose all the terms that apply to this matrix.</p>@
qu.7.5.answer=1, 3, 5, 6, 10, 11@
qu.7.5.choice.1=invertible@
qu.7.5.choice.2=identity@
qu.7.5.choice.3=full rank@
qu.7.5.choice.4=idempotent@
qu.7.5.choice.5=symmetric@
qu.7.5.choice.6=triangular@
qu.7.5.choice.7=zero matrix@
qu.7.5.choice.8=negative definite@
qu.7.5.choice.9=indefinite@
qu.7.5.choice.10=diagonal matrix@
qu.7.5.choice.11=positive definite@
qu.7.5.fixed=@

qu.7.6.mode=Multiple Selection@
qu.7.6.name=Matrix types -- idempotent@
qu.7.6.comment=@
qu.7.6.editing=useHTML@
qu.7.6.solution=@
qu.7.6.algorithm=$a=range(1,9,1);
$b=range(1,9,1);
$n=range(1,4,1);
$k=range(0,2);
$C=maple("
B:=(Matrix(2,2,[[0,-1],[1,0]]))^($n):
C1:=Matrix([[1,0,0],[0,B[1,1],B[1,2]],[0,B[2,1],B[2,2]]]):
C2:=Matrix([[B[1,1],0,B[1,2]],[0,1,0],[B[2,1],0,B[2,2]]]):
C3:=Matrix([[B[1,1],B[1,2],0],[B[2,1],B[2,2],0],[0,0,1]]):
convert(C1,string),convert(C2,string),convert(C3,string)
");
$C0=switch(($k),$C);
$A=maple("
A0:=Matrix([[($a),(-$a),0],[($a-1),(1-$a),0],[0,0,1]]):
A1:=($C0).A0.(LinearAlgebra[MatrixInverse]($C0)):
A2:=MathML[ExportPresentation](A1):

A2
");
$Q=switch(0,$A);@
qu.7.6.uid=f6cc8d9d-663f-45e6-aa05-0000b0876e40@
qu.7.6.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Types;
  Sub-Topic=Idempotent Matrix;
  Author=Asha Sadanand;
  Difficulty=Hard;
  Feature=Multiple Select;
@
qu.7.6.question=<p>Consider the following matrix.</p>
<p>&nbsp;$Q</p>
<p>Choose all the terms that apply to this matrix.</p>@
qu.7.6.answer=4@
qu.7.6.choice.1=invertible@
qu.7.6.choice.2=identity@
qu.7.6.choice.3=full rank@
qu.7.6.choice.4=idempotent@
qu.7.6.choice.5=symmetric@
qu.7.6.choice.6=triangular@
qu.7.6.choice.7=zero matrix@
qu.7.6.choice.8=negative definite@
qu.7.6.choice.9=indefinite@
qu.7.6.choice.10=diagonal matrix@
qu.7.6.choice.11=positive definite@
qu.7.6.fixed=@

qu.7.7.mode=Multiple Selection@
qu.7.7.name=Matrix types -- zero matrix@
qu.7.7.comment=@
qu.7.7.editing=useHTML@
qu.7.7.solution=@
qu.7.7.algorithm=$A=maple("
A1:=Matrix([[0,0,0],[0,0,0],[0,0,0]]):
A2:=MathML[ExportPresentation](A1):
A2
");
$Q=switch(0,$A);@
qu.7.7.uid=82cb25f2-7325-4066-bf86-07e5318ec82e@
qu.7.7.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Types;
  Sub-Topic=Zero Matrix;
  Author=Asha Sadanand;
  Difficulty=Easy;
  Feature=Multiple Select;
@
qu.7.7.question=<p>Consider the following matrix.</p>
<p>&nbsp;$Q</p>
<p>Choose all the terms that apply to this matrix.</p>@
qu.7.7.answer=5, 6, 7@
qu.7.7.choice.1=invertible@
qu.7.7.choice.2=identity@
qu.7.7.choice.3=full rank@
qu.7.7.choice.4=idempotent@
qu.7.7.choice.5=symmetric@
qu.7.7.choice.6=triangular@
qu.7.7.choice.7=zero matrix@
qu.7.7.choice.8=negative definite@
qu.7.7.choice.9=indefinite@
qu.7.7.choice.10=diagonal matrix@
qu.7.7.choice.11=positive definite@
qu.7.7.fixed=@

qu.7.8.mode=Multiple Selection@
qu.7.8.name=Matrix types -- identity matrix@
qu.7.8.comment=@
qu.7.8.editing=useHTML@
qu.7.8.solution=@
qu.7.8.algorithm=$A=maple("
A1:=Matrix([[1,0,0],[0,1,0],[0,0,1]]):
A2:=MathML[ExportPresentation](A1):
A2
");
$Q=switch(0,$A);@
qu.7.8.uid=67fb3a04-64d3-405c-a420-ac274593ee12@
qu.7.8.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Types;
  Sub-Topic=Identity Matrix;
  Author=Asha Sadanand;
  Difficulty=Easy;
  Feature=Multiple Select;
@
qu.7.8.question=<p>Consider the following matrix.</p>
<p>&nbsp;$Q</p>
<p>Choose all the terms that apply to this matrix.</p>@
qu.7.8.answer=1, 2, 3, 4, 5, 6, 10, 11@
qu.7.8.choice.1=invertible@
qu.7.8.choice.2=identity@
qu.7.8.choice.3=full rank@
qu.7.8.choice.4=idempotent@
qu.7.8.choice.5=symmetric@
qu.7.8.choice.6=triangular@
qu.7.8.choice.7=zero matrix@
qu.7.8.choice.8=negative definite@
qu.7.8.choice.9=indefinite@
qu.7.8.choice.10=diagonal matrix@
qu.7.8.choice.11=positive definite@
qu.7.8.fixed=@

qu.7.9.mode=Multiple Selection@
qu.7.9.name=Matrix types - non-diagonal negative definite@
qu.7.9.comment=@
qu.7.9.editing=useHTML@
qu.7.9.solution=@
qu.7.9.algorithm=$a=-1*range(1,9,1);
$b=-1*range(1,9,1);
$c=-1*range(1,9,1);
$A=maple("
m1:=LinearAlgebra[RandomMatrix](3,generator=rand(1..3),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m1) = 0 do m1 := LinearAlgebra[RandomMatrix](3, generator = rand(1 .. 3), attributes = [nonsingular]) end do:
m2:=LinearAlgebra[Determinant](m1):
m5:=LinearAlgebra[MatrixInverse](m1):
A0:=Matrix([[$a,0,0],[0,$b,0],[0,0,$c]]):
A1:=m2*(m5.A0.m1):
A2:=MathML[ExportPresentation](A1):
A2
");
$Q=switch(0,$A);@
qu.7.9.uid=e157b650-360c-4d41-9504-b8753c191498@
qu.7.9.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Types;
  Sub-Topic=Non-Diagonal Negative Definite;
  Author=Asha Sadanand;
  Difficulty=Easy;
  Feature=Multiple Selection;
@
qu.7.9.question=<p>Consider the following matrix.</p>
<p>&nbsp;$Q</p>
<p>Choose all the terms that apply to this matrix.</p>@
qu.7.9.answer=1, 3, 8@
qu.7.9.choice.1=invertible@
qu.7.9.choice.2=identity@
qu.7.9.choice.3=full rank@
qu.7.9.choice.4=idempotent@
qu.7.9.choice.5=symmetric@
qu.7.9.choice.6=triangular@
qu.7.9.choice.7=zero matrix@
qu.7.9.choice.8=negative definite@
qu.7.9.choice.9=indefinite@
qu.7.9.choice.10=diagonal matrix@
qu.7.9.choice.11=positive definite@
qu.7.9.fixed=@

