College Algebra (6th Edition)

Published by Pearson
ISBN 10: 0-32178-228-3
ISBN 13: 978-0-32178-228-1

Chapter 6 - Matrices and Determinants - Exercise Set 6.4 - Page 640: 57

Answer

If A is not a square matrix, and B is the multiplicative inverse of A, then the products AB and BA do not result in the same multiplicative identity matrix.

Work Step by Step

Let A be an m$\times$n matrix. In order for B to be the multiplicative inverse of A B must an n$\times$m matrix, because AB and BA should equal each other, and they should equal the same multiplicative identity matrix, which is a square matrix $AB$ (order: m$\times$m) and $BA $(n$\times$n) should have equal order, that is, $I_{m}=I_{n}.$ Therefore, m=n. So, A must be a square matrix.
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