. . Gauss-Jordan Method Let A be a non-singular square matrix of order n . Let B be the inverse of A .
Then, we have AB BA I n . By the property of I n , we have A I A AI Consider the equation A = I A …( ) Since A is non-singular, pre-multiplying by a sequence of elementary matrices (row operations) on both sides of ( ), A on the left-hand-side of ( ) is transformed to the identity matrix I n and the same sequence of elementary matrices (row operations) transforms I n of the right-hand-side of ( ) to a matrix B . So, equation ( ) transforms to I BA n = .Hence, the inverse of A is B . That is, A B − = + − + - - Note If E E E k are elementary matrices (row operations) such that E E E I , then E E E − = Transforming a non-singular matrix A to the form I n by applying elementary row operations, is called Gauss-Jordan method .
The steps in finding A − by Gauss-Jordan method are given below: Step Augment the identity matrix I n on the right-side of A to get the matrix A I n ] . Step Obtain elementary matrices (row operations) E E E k such that E E E I Apply E E E k on A I n