Question: if ab is invertible then both a and b are invertible?
Answer: It follows that A = B-1, hence B = A-1. Theorem A square matrix A is invertible if and only if x = 0 is the only solution of the matrix equation Ax = 0. ... If the product AB is invertible, then both A and B are invertible. Proof: Let C = B(AB)-1 and D = (AB)-1A.