If We know the determinants of A and B but suppose that we l

If We know the determinants of A and B, but suppose that we lost our original matrices A and B. Which of the determinants ( A + B, A - B, AB, A^-1, B^T) will we still be able to recover, even without having A or B at hand? Explain your reasoning.

Solution

We know

det(AB)=det(A)det(B)

So we can find determinant of AB

det(AA^{-1})=det(I)=1=det(A)det(A^{-1})

So,det(A^{-1})=1/det(A)

det(B^T)=det(B)