## Precalculus (6th Edition) Blitzer

Published by Pearson

# Chapter 8 - Section 8.4 - Multiplicative Inverses of Matrices and Matrix Equations - Exercise Set - Page 933: 57

#### Answer

Non-square matrices cannot have a multiplicative inverse since their products are not equal to the same multiplicative identity matrix.

#### Work Step by Step

Suppose there is an $n\times n$ square matrix A. If there exists another $n\times n$ matrix ${{A}^{-1}}$ such that: \begin{align} & A{{A}^{-1}}={{I}_{n}} \\ & {{A}^{-1}}A={{I}_{n}} \end{align} Then, the square matrix ${{A}^{-1}}$ is said to be a multiplicative inverse of the square matrix A. The result of multiplication of matrix A and ${{A}^{-1}}$ is an $n\times n$ square matrix, which is an identity matrix ${{I}_{n}}$. Only square matrices can have multiplicative inverses. And also, it is not necessary that every square matrix will possess an inverse matrix. Non-square matrices cannot have a multiplicative inverse. This is because of the following reason: A non-square matrix is a matrix that has a different number of rows and columns. If A is a $m\times n$ matrix and B is a $n\times m$ matrix ( $n\ne m$ ), then the products AB and BA will be of different orders. This means two products AB and BA are not equal to the same multiplicative identity matrix.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.