Chapter 8 - Section 8.3 - Matrix Operations and Their Applications - Exercise Set - Page 918: 36

a) $AB=\left[ \begin{matrix} 18 & 1 \\ -1 & 15 \\ \end{matrix} \right]$ b) $BA=\left[ \begin{matrix} 0 & 1 & -7 & 0 \\ 3 & -1 & 1 & 3 \\ 2 & -3 & 17 & 2 \\ 17 & -6 & 8 & 17 \\ \end{matrix} \right]$

(a) Consider, $\begin{align} & AB=\left[ \begin{array}{*{35}{l}} 2 & -1 & 3 & 2 \\ 1 & 0 & -2 & 1 \\ \end{array} \right]\left[ \begin{array}{*{35}{l}} -1 & 2 \\ 1 & 1 \\ 3 & -4 \\ 6 & 5 \\ \end{array} \right] \\ & =\left[ \begin{matrix} 2\left( -1 \right)-1\left( 1 \right)+3\left( 3 \right)+2\left( 6 \right) & 2\left( 2 \right)-1\left( 1 \right)+3\left( -4 \right)+2\left( 5 \right) \\ 1\left( -1 \right)+0\left( 1 \right)-2\left( 3 \right)+1\left( 6 \right) & 1\left( 2 \right)+0\left( 1 \right)-2\left( -4 \right)+1\left( 5 \right) \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} -2-1+9+12 & 4-1-12+10 \\ -1+0-6+6 & 2+0+8+5 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 18 & 1 \\ -1 & 15 \\ \end{matrix} \right] \end{align}$ (b) Consider, $\begin{align} & BA=\left[ \begin{array}{*{35}{l}} -1 & 2 \\ 1 & 1 \\ 3 & -4 \\ 6 & 5 \\ \end{array} \right]\left[ \begin{array}{*{35}{l}} 2 & -1 & 3 & 2 \\ 1 & 0 & -2 & 1 \\ \end{array} \right] \\ & =\left[ \begin{matrix} -1\left( 2 \right)+2\left( 1 \right) & -1\left( -1 \right)+2\left( 0 \right) & -1\left( 3 \right)+2\left( -2 \right) & -1\left( 2 \right)+2\left( 1 \right) \\ 1\left( 2 \right)+1\left( 1 \right) & 1\left( -1 \right)+1\left( 0 \right) & 1\left( 3 \right)+1\left( -2 \right) & 1\left( 2 \right)+1\left( 1 \right) \\ 3\left( 2 \right)-4\left( 1 \right) & 3\left( -1 \right)-4\left( 0 \right) & 3\left( 3 \right)-4\left( -2 \right) & 3\left( 2 \right)-4\left( 1 \right) \\ 6\left( 2 \right)+5\left( 1 \right) & 6\left( -1 \right)+5\left( 0 \right) & 6\left( 3 \right)+5\left( -2 \right) & 6\left( 2 \right)+5\left( 1 \right) \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} -2+2 & 1+0 & -3-4 & -2+2 \\ 2+1 & -1+0 & 3-2 & 2+1 \\ 6-4 & -3+0 & 9+8 & 6-4 \\ 12+5 & -6+0 & 18-10 & 12+5 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 0 & 1 & -7 & 0 \\ 3 & -1 & 1 & 3 \\ 2 & -3 & 17 & 2 \\ 17 & -6 & 8 & 17 \\ \end{matrix} \right] \end{align}$