@Carl Love

i have tried eliminiate but no solution return

m5 := solve([AA[1,1]=0,AA[1,2]=0,AA[2,1]=0,AA[2,2]=0, a= Jesus2[1,1], b=Jesus2[1,2], c=Jesus2[2,1], d=Jesus2[2,2], a*b = 1, c*d = 1, conjugate(a*c)-b*d = 0,a*d-b*c=sqrt(3)*I],[a,b,c,d]);

Original question is how to solve a, b, c and d for below equations:

Both known Matrix and solution Matrix Properties

Matrix([[a,b],[c,d]]);

a*b = 1

c*d = 1

conjugate(a*c)-b*d = 0

Determinant(Matrix([[a,b],[c,d]])) = sqrt(3)*I

a*d-b*c = sqrt(3)*I

How to solve system AA when known Jesus7 to find Jesus2 which is Matrix([[a,b],[c,d]]) ?

Jesus7 := Matrix(2, 2, {(1, 1) = (1296309/5)*(22243/5+((7358/5)*I)*sqrt(3))/((-384813/10+((33701837/10)*I)*sqrt(3))*(-2277/10+((13/10)*I)*sqrt(3))), (1, 2) = (1296309/5)*(22243/5-((7358/5)*I)*sqrt(3))/((-384813/10-((33701837/10)*I)*sqrt(3))*(-2277/10-((13/10)*I)*sqrt(3))), (2, 1) = -(1/5)*(-25372581+(8329256*I)*sqrt(3))/(-384813/10+((33701837/10)*I)*sqrt(3)), (2, 2) = -(1/5)*(-25372581-(8329256*I)*sqrt(3))/(-384813/10-((33701837/10)*I)*sqrt(3))});

AA := simplify(MatrixMatrixMultiply(Matrix([[a,b],[c,d]]),Jesus7)- MatrixMatrixMultiply(Jesus7,Matrix([[a,b],[c,d]]))):

I am using different properties and even full properties , still can not find the solution equal to exact solution, but number of equations had already been more than number of variables? where is incorrect?

m5 := solve([AA[1,1]=0,AA[1,2]=0,AA[2,1]=0,AA[2,2]=0, a*b = 1, c*d = 1, conjugate(a*c)-b*d = 0,a*d-b*c=sqrt(3)*I],[a,b,c,d]);

m5 := solve([AA[1,1]=0,AA[1,2]=0,AA[2,1]=0,AA[2,2]=0, a*b = 1, c*d = 1, a*d-b*c=sqrt(3)*I],[a,b,c,d]);

m5 := solve([AA[1,1]=0,AA[1,2]=0,AA[2,1]=0,AA[2,2]=0, a*b = 1, c*d = 1, conjugate(a*c)-b*d = 0],[a,b,c,d]);

m5 := solve([AA[1,1]=0,AA[1,2]=0,AA[2,1]=0,AA[2,2]=0, a*b = 1, c*d = 1],[a,b,c,d]);

Solution: Matrix([[a,b],[c,d]]) should be

Jesus2 := [[1313319/2 (3439+1145 I sqrt(3))/((-2292+1313317 I sqrt(3)) (-1146+I sqrt(3))),1313319/2 (3439-1145 I sqrt(3))/((-2292-1313317 I sqrt(3)) (-1146-I sqrt(3)))],[-1/2 (-3942243+1311025 I sqrt(3))/(-2292+1313317 I sqrt(3)),-1/2 (-3942243-1311025 I sqrt(3))/(-2292-1313317 I sqrt(3))]];

i find this combination of equations has solution

m5 := solve([AA[1,1]=0,AA[1,2]=0,AA[2,1]=0,AA[2,2]=0, a*b = 1, c*d = 1, a*d-b*c=sqrt(3)*I],[a,b,c,d]);

but the solution is not Jesus2, i did not know what inside in solve function, it is like somewhere has evalf it and then change back to fraction, may be my guess is incorrect.