## Inverse and the Gauss-Jordan Method

We first give a consequence of Theorem 2.5.8 and then use it to find the inverse of an invertible matrix.

COROLLARY 2.5.13   Let be an invertible matrix. Suppose that a sequence of elementary row-operations reduces to the identity matrix. Then the same sequence of elementary row-operations when applied to the identity matrix yields

Proof. Let be a square matrix of order Also, let be a sequence of elementary row operations such that Then This implies height6pt width 6pt depth 0pt

Summary: Let be an matrix. Apply the Gauss-Jordan method to the matrix Suppose the row reduced echelon form of the matrix is If then or else is not invertible.

EXAMPLE 2.5.14   Find the inverse of the matrix using the Gauss-Jordan method.

Solution: Consider the matrix A sequence of steps in the Gauss-Jordan method are:

1. Thus, the inverse of the given matrix is

EXERCISE 2.5.15   Find the inverse of the following matrices using the Gauss-Jordan method.

A K Lal 2007-09-12