We recall that a square matrix A is nonsingular if it has a left inverse B such that . It is then shown that B, if exists, is unique and also a right inverse, i.e., . Therefore, we call B the inverse of A and denote it by .
If matrix A is nonsingular, then and
Using the product formula (5.15), we have
Theorem 5.12 shows that is a necessary condition for A to be nonsingular. Later, in Theorem 5.16, it will be shown that it is also a sufficient condition, i.e., implies the existence of .