Page 68 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

DEFINITION

If A is a square matrix, and if a matrix B of the same size can be found such that , then A is said to be invertible

and B is called an inverse of A. If no such matrix B can be found, then A is said to be singular.

EXAMPLE 5 Verifying the Inverse Requirements
The matrix
since
and

EXAMPLE 6 A Matrix with No Inverse
The matrix
is singular. To see why, let

be any matrix. The third column of is

Thus

Properties of Inverses

It is reasonable to ask whether an invertible matrix can have more than one inverse. The next theorem shows that the answer is