Chapter 8 - Section 8.3 - Matrix Operations and Their Applications - Exercise Set - Page 918: 35

a) $AB=\left[ \begin{matrix} 0 & 0 \\ 0 & 0 \\ \end{matrix} \right]$ b) $BA=\left[ \begin{matrix} 4 & -1 & -3 & 1 \\ -1 & 4 & -3 & 2 \\ 14 & -11 & -3 & -1 \\ 25 & -25 & 0 & -5 \\ \end{matrix} \right]$

(a) Consider, $\begin{align} & AB=\left[ \begin{array}{*{35}{l}} 2 & -3 & 1 & -1 \\ 1 & 1 & -2 & 1 \\ \end{array} \right]\left[ \begin{array}{*{35}{l}} 1 & 2 \\ -1 & 1 \\ 5 & 4 \\ 10 & 5 \\ \end{array} \right] \\ & =\left[ \begin{matrix} 2\left( 1 \right)-3\left( -1 \right)+1\left( 5 \right)-1\left( 10 \right) & 2\left( 2 \right)-3\left( 1 \right)+1\left( 4 \right)-1\left( 5 \right) \\ 1\left( 1 \right)+1\left( -1 \right)-2\left( 5 \right)+1\left( 10 \right) & 1\left( 2 \right)+1\left( 1 \right)-2\left( 4 \right)+1\left( 5 \right) \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 2+3+5-10 & 4-3+4-5 \\ 1-1-10+10 & 2+1-8+5 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 0 & 0 \\ 0 & 0 \\ \end{matrix} \right] \end{align}$ (b) Consider, $\begin{align} & BA=\left[ \begin{array}{*{35}{l}} 1 & 2 \\ -1 & 1 \\ 5 & 4 \\ 10 & 5 \\ \end{array} \right]\left[ \begin{array}{*{35}{l}} 2 & -3 & 1 & -1 \\ 1 & 1 & -2 & 1 \\ \end{array} \right] \\ & =\left[ \begin{matrix} 1\left( 2 \right)+2\left( 1 \right) & 1\left( -3 \right)+2\left( 1 \right) & 1\left( 1 \right)+2\left( -2 \right) & 1\left( -1 \right)+2\left( 1 \right) \\ -1\left( 2 \right)+1\left( 1 \right) & -1\left( -3 \right)+1\left( 1 \right) & -1\left( 1 \right)+1\left( -2 \right) & -1\left( -1 \right)+1\left( 1 \right) \\ 5\left( 2 \right)+4\left( 1 \right) & 5\left( -3 \right)+4\left( 1 \right) & 5\left( 1 \right)+4\left( -2 \right) & 5\left( -1 \right)+4\left( 1 \right) \\ 10\left( 2 \right)+5\left( 1 \right) & 10\left( -3 \right)+5\left( 1 \right) & 10\left( 1 \right)+5\left( -2 \right) & 10\left( -1 \right)+5\left( 1 \right) \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 2+2 & -3+2 & 1-4 & -1+2 \\ -2+1 & 3+1 & -1-2 & 1+1 \\ 10+4 & -15+4 & 5-8 & -5+4 \\ 20+5 & -30+5 & 10-10 & -10+5 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 4 & -1 & -3 & 1 \\ -1 & 4 & -3 & 2 \\ 14 & -11 & -3 & -1 \\ 25 & -25 & 0 & -5 \\ \end{matrix} \right] \end{align}$