Inverse-matrix verification test ($AC=I$)

A claimed inverse matrix C for a finite square matrix A must satisfy $AC=CA=I$. A single product entry differing from the identity matrix disproves the claim. Conversely, in finite dimension one exact identity such as $AC=I$ already implies that A has full rank, so A is invertible and $C=A^{-1}$; the other product then equals I as well. Checking the product is useful before using a supplied inverse in a similarity transformation.