Chapter 8 - Section 8.3 - Matrix Operations and Their Applications - Exercise Set - Page 918: 39

$BC+CB=\left[ \begin{matrix} 11 & -1 \\ -7 & -3 \\ \end{matrix} \right]$

First we will find the product matrix $BC$ as follows, $\begin{align} & BC=\left[ \begin{matrix} 5 & 1 \\ -2 & -2 \\ \end{matrix} \right]\left[ \begin{matrix} 1 & -1 \\ -1 & 1 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 5\left( 1 \right)+1\left( -1 \right) & 5\left( -1 \right)+1\left( 1 \right) \\ -2\left( 1 \right)-2\left( -1 \right) & -2\left( -1 \right)-2\left( 1 \right) \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 5-1 & -5+1 \\ -2+2 & 2-2 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 4 & -4 \\ 0 & 0 \\ \end{matrix} \right] \end{align}$ Now we will find $CB$ as follows: $\begin{align} & CB=\left[ \begin{matrix} 1 & -1 \\ -1 & 1 \\ \end{matrix} \right]\left[ \begin{matrix} 5 & 1 \\ -2 & -2 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 1\left( 5 \right)-1\left( -2 \right) & 1\left( 1 \right)-1\left( -2 \right) \\ -1\left( 5 \right)+1\left( -2 \right) & -1\left( 1 \right)+1\left( -2 \right) \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 5+2 & 1+2 \\ -5-2 & -1-2 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 7 & 3 \\ -7 & -3 \\ \end{matrix} \right] \end{align}$ Now we will add the matrices $BC\text{ and }CB$ as follows: $\begin{align} & BC+CB=\left[ \begin{matrix} 4 & -4 \\ 0 & 0 \\ \end{matrix} \right]+\left[ \begin{matrix} 7 & 3 \\ -7 & -3 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 4+7 & -4+3 \\ 0-7 & 0-3 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 11 & -1 \\ -7 & -3 \\ \end{matrix} \right] \end{align}$