## Precalculus (6th Edition) Blitzer

a) $AB=\left[ \begin{matrix} 2 & -8 & 20 \\ 8 & 3 & 5 \\ 10 & 2 & 10 \\ \end{matrix} \right]$ b) $BA=\left[ \begin{matrix} 12 & 14 \\ 9 & 3 \\ \end{matrix} \right]$
(a) Consider, \begin{align} & AB=\left[ \begin{array}{*{35}{l}} 2 & 4 \\ 3 & 1 \\ 4 & 2 \\ \end{array} \right]\left[ \begin{array}{*{35}{l}} 3 & 2 & 0 \\ -1 & -3 & 5 \\ \end{array} \right] \\ & =\left[ \begin{matrix} 2\left( 3 \right)+4\left( -1 \right) & 2\left( 2 \right)+4\left( -3 \right) & 2\left( 0 \right)+4\left( 5 \right) \\ 3\left( 3 \right)+1\left( -1 \right) & 3\left( 2 \right)+1\left( -3 \right) & 3\left( 0 \right)+1\left( 5 \right) \\ 4\left( 3 \right)+2\left( -1 \right) & 4\left( 2 \right)+2\left( -3 \right) & 4\left( 0 \right)+2\left( 5 \right) \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 6-4 & 4-12 & 0+20 \\ 9-1 & 6-3 & 0+5 \\ 12-2 & 8-6 & 0+10 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 2 & -8 & 20 \\ 8 & 3 & 5 \\ 10 & 2 & 10 \\ \end{matrix} \right] \end{align} (b) \begin{align} & BA=\left[ \begin{array}{*{35}{l}} 3 & 2 & 0 \\ -1 & -3 & 5 \\ \end{array} \right]\left[ \begin{array}{*{35}{l}} 2 & 4 \\ 3 & 1 \\ 4 & 2 \\ \end{array} \right] \\ & =\left[ \begin{matrix} 3\left( 2 \right)+2\left( 3 \right)+0\left( 4 \right) & 3\left( 4 \right)+2\left( 1 \right)+0\left( 2 \right) \\ -1\left( 2 \right)-3\left( 3 \right)+5\left( 4 \right) & -1\left( 4 \right)-3\left( 1 \right)+5\left( 2 \right) \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 6+6+0 & 12+2+0 \\ -2-9+20 & -4-3+10 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 12 & 14 \\ 9 & 3 \\ \end{matrix} \right] \end{align}