## Precalculus (6th Edition) Blitzer

Published by Pearson

# Chapter 8 - Review Exercises - Page 951: 37

#### Answer

Matrix $B$ is not a multiplicative inverse of the matrix $A$.

#### Work Step by Step

Find the product of $AB$ as follows: \begin{align} & AB=\left[ \begin{matrix} 2 & 7 \\ 1 & 4 \\ \end{matrix} \right]\left[ \begin{matrix} 4 & -7 \\ -1 & 3 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 2\left( 4 \right)+7\left( -1 \right) & 2\left( -7 \right)+7\left( 3 \right) \\ 1\left( 4 \right)+4\left( -1 \right) & 1\left( -7 \right)+4\left( 3 \right) \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 8-7 & -14+21 \\ 4-4 & -7+12 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 1 & 7 \\ 0 & 5 \\ \end{matrix} \right] \end{align} Therefore, $AB=\left[ \begin{matrix} 1 & 7 \\ 0 & 5 \\ \end{matrix} \right]$ Next we will find the product of $BA$ as follows: \begin{align} & BA=\left[ \begin{matrix} 4 & -7 \\ -1 & 3 \\ \end{matrix} \right]\left[ \begin{matrix} 2 & 7 \\ 1 & 4 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 4\left( 2 \right)-7\left( 1 \right) & 4\left( 7 \right)-7\left( 4 \right) \\ -1\left( 2 \right)+3\left( 1 \right) & -1\left( 7 \right)+3\left( 4 \right) \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 8-7 & 28-28 \\ -2+3 & -7+12 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 1 & 0 \\ 1 & 5 \\ \end{matrix} \right] \end{align} As $AB\ne BA\ne I$, where x is an identity matrix, therefore, the given matrix is not a multiplicative inverse of the matrix .

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