1. The problem statement, all variables and given/known data Let A and B be nxn matrices such that AB is invertible. Show that A and B are also invertible. 2. Relevant equations 3. The attempt at a solution I feel that there are many ways to do this. My first idea was to use the fact that CAB = I and ABC = I for some C, which implies that (CA)B = I and A(BC) = I, which means that A has a right inverse and B has a left inverse. But since A and B are nxn, BC and CA are not just right and left inverses, respectively, but they are the inverses of A and B. Is this correct reasoning? Also, would it be any better to make use of the isomorphism from matrices and linear transformations, prove the result for linear transformations, and hence prove the result for matrices?