Suppose A and B are two square matrices of the same size Sho

Suppose A and B are two square matrices of the same size. Show that AB is invertible if and only if A and B are both invertible.

Solution

Let A and B be n x n matrices.

Let AB be invertible. Then det (AB) 0 . Since det (AB) = det(A) det (B), it means that det(A) 0 and det (B) 0 which implies that both A and B are invertible.

Let A and B be both invertible. Then det(A) 0 and det (B) 0 so that det(A) det (B) 0. This implies that det (AB) 0 [ as det (AB) = det(A) det (B) ]. This implies that AB is invertible.