Question

Let A and B be n×n matrices. Determine whether |A + AB| = |A + BA|.

Answer 1

Yes, they're equal.

det(A + AB) = det(AI + AB) = det(A (I + B)) = det(A) det(I + B)

and

det(A + BA) = det(IA + BA) = det((I + B) A) = det(I + B) det(A)

which are the same.