EXAMPLE 6 We can use Gaussian elimination to find a right inverse of an m xn matrix A, so long as the rank of A is equal to m. The fact that we have free variables when m < n will give many choices of right inverse. For example, taking 1. A = -1 2 -1 we apply Gaussian elimination to the augmented matrix 1 -1 1 1 -1 1 1 0. 2 -1 0. 0. 0. 1 -2 -2 1 0- 1 -1 1 -2 -2 []- From this we see that the general solution of Ax = is -1 X = +s is and the general solution of Ax = X = + If we take s =t = 0, we get the right inverse |-1 B = -2 1 0. 0.

EXAMPLE 6 We can use Gaussian elimination to find a right inverse of an m xn matrix A, so long as the rank of A is equal to m. The fact that we have free variables when m < n will give many choices of right inverse. For example, taking 1. A = -1 2 -1 we apply Gaussian elimination to the augmented matrix 1 -1 1 1 -1 1 1 0. 2 -1 0. 0. 0. 1 -2 -2 1 0- 1 -1 1 -2 -2 []- From this we see that the general solution of Ax = is -1 X = +s is and the general solution of Ax = X = + If we take s =t = 0, we get the right inverse |-1 B = -2 1 0. 0.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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