1. The problem statement, all variables and given/known data Let A and B be n by n matrices such that A is invertible and B is not invertible. Then, AB is not invertible. 2. Relevant equations 3. The attempt at a solution We know that A is invertible, so there exists a matrix C such that CA = I. Then we can right -multiply by B so that CAB = IB = I. Then by the associative property C(AB) = I. By the same argument, we can show that there is a C such that (AB)C = I. So AB has an inverse. Obviously this is wrong, because in order for AB to have an inverse, both A and B must have an inverse. So what am I doing wrong?