In 2d math mode that you use, the easiest way to calculate the determinant is to use two vertical bars, which you can enter from the keyboard:

restart;
u:=u0(x,y)+z*u1(x,y):
v:=v0(x,y)+z*v1(x,y):
A:=<a,b; c,d>;
B:=<diff(u,x), diff(v,y)>;
C:=A.B;

General approach in applying to your example:

restart; 
F := ((1/2)*a[2]-a[1]+(1/2)*a[0])*n^2+(-(1/2)*a[2]+2*a[1]-(3/2)*a[0])*n+a[0]; 
N := 3; 
var := [seq(a[n-1], n = 1 .. N)]; 
factor~([seq(op(k, foldl(collect, F, var[]))/var[N-k+1], k = 1 .. N)]);

This is a simple arithmetic:

480*1/(5+1+1/4);
                                      384/5

So the answer is $76.8

A:=<0,1,1,1,0,1; 1,0,0,0,1,1; 1,0,0,0,1,1; 1,0,0,0,1,1;0,1,1,1,0,1; 1,1,1,1,1,0>:
evalc(LinearAlgebra:-Eigenvalues(A));  # Symbolic (exact) result
evalf(%);  # Approximate result
                             

L:=[[TC,DB], [], [TD,JK], [IW,CM], [], [KJ,DJ]];
remove(s->s=[], L);  # or
remove(`=`, L, []);

The first equation is set incorrectly. See the corrected file  SimEquations_new.mw

First we inserted the row  R  into the matrix  A, and then inserted  the column  C  into the matrix  A1:

A:=LinearAlgebra:-RandomMatrix(6, generator=0..9);
R:=LinearAlgebra:-RandomVector[row](6, generator=0..9);
C:=LinearAlgebra:-RandomVector(7, generator=0..9);
A1:=<<A[..3], R>, A[4..]>;
A2:=<A1[..,..3] | C | A1 [..,4..]>;

Edit.

To plot space curves, I usually use  plots:-spacecurve  command, because it gives the best quality, and instead of transparency I select the appropriate  orientation:

restart;
PP:=0.8707945038*exp(-50.00000000*(m-.842e-1)^2+2.745342070*(m-.842e-1)*(a-2.3722)-.1046792095*(a-2.3722)^2):
A:=plots:-spacecurve([a, -0.2, eval(PP,m=-0.2)+0.005], a = -5 .. 8, color=red, thickness=4):
B:=plot3d(PP, a = -5 .. 8, m = -0.7 .. 0.7, style=surface, color=khaki, numpoints=5000):
C:=plot3d([a, -0.2, z], a = -5 .. 8, z = 0 .. 0.9, style=surface, color="LightBlue"):
plots:-display(A,B,C, orientation=[-30,70]);

Edit.

In this line  k = 10; Matrix([[time, harmonic], seq([t, evalf(S2)], t = 0 .. 1, 0.001)]);

of your code, the matrix is not calculated, because should be   k:=10  instead of  k=10

In further lines, if  k  will take different values, you must first execute  k:='k'

Under the summation sign write  'k'=any value

X:=<3,4,2,5>:  Y:=<5,3,0,1>:  Z:=<1,2,3,4>:
plots:-display(plot(Z,X, color=red), plot(Z,Y, color=blue));  # or
plot(convert~([`[]`~(Z,X),`[]`~(Z,Y)],list));  # or 
plot(map(convert,[zip(`[]`,Z,X),zip(`[]`,Z,Y)],list));  # for older versions of Maple
 

restart:    
# a piecewise function    
p1:=piecewise(a(t)^2=0,<cos(a(t))^2+sin(a(t))^2,0>,<1,1/a(t)>):
evalindets(p1, Vector, p->diff~(p,t));

P:=[`TC,DB`, `PC,JL`, `TD,JK`, `IW,CM`, `CC,PG`, `KJ,DJ`];
map(`[]`, P);

See  untitled5new.mw

You wrote  " v := x->% " equivalent to "v:=unapply(%,x)" . It is false. The main difference between these two ways of specifying a function is that when you write  f:=t->A  in the input line, Maple does nothing, by this simply a procedure is specified, which calculates something only when it is called.  f:=unapply(A, t)  also sets a procedure, but at the same time, at the time of its assignment, Maple calculates what is written in  A .

Compare:
t^2;
f:=t->%;
5;
f(2);

and

t^2;
g:=unapply(%, t);
5;
g(2);