Solve this Linear Algebra problem please Show ALL work Thank

Solve this Linear Algebra problem please. Show ALL work. Thank you!

If A and B are invertible and of the same size, express det((AB)^-1) in terms of det(A) and det(B).

Solution

Given det((AB)^-1) = det(B^-1A^-1) [ since (AB)^-1 = B^-1A^-1

= det(B^-1)det(A^-1) [ since det(AB) = det(A)det(B)   

= 1/detB*1/det(A) [ since det(A^-1) = 1/detA

= 1/[det(A)det(B) ]