## Precalculus (6th Edition) Blitzer

The value of $AB-4A$ is $\left[ \begin{matrix} -2 & 10 \\ -5 & 7 \\ 15 & -15 \\ \end{matrix} \right]$.
We have to calculate the product of AB as given below: \begin{align} & AB=\left[ \begin{matrix} 4 & 2 \\ 1 & -1 \\ 0 & 5 \\ \end{matrix} \right]\left[ \begin{matrix} 2 & 4 \\ 3 & 1 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 4\cdot 2+2\cdot 3 & 4\cdot 4+2\cdot 1 \\ 1\cdot 2+\left( -1 \right)\cdot 3 & 1\cdot 4+\left( -1 \right)\cdot 1 \\ 0\cdot 2+5\cdot 3 & 0\cdot 4+5\cdot 1 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 8+6 & 16+2 \\ 2-3 & 4-1 \\ 0+15 & 0+5 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 14 & 18 \\ -1 & 3 \\ 15 & 5 \\ \end{matrix} \right] \end{align} Then, for the value of $4A$ is: \begin{align} & 4A=4\left[ \begin{matrix} 4 & 2 \\ 1 & -1 \\ 0 & 5 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 4\cdot 4 & 4\cdot 2 \\ 4\cdot 1 & 4\cdot \left( -1 \right) \\ 4\cdot 0 & 4\cdot 5 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 16 & 8 \\ 4 & -4 \\ 0 & 20 \\ \end{matrix} \right] \end{align} Next, for the value of the given expression $AB-4A$: \begin{align} & AB-4A=\left[ \begin{matrix} 14 & 18 \\ -1 & 3 \\ 15 & 5 \\ \end{matrix} \right]-\left[ \begin{matrix} 16 & 8 \\ 4 & -4 \\ 0 & 20 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 14-16 & 18-8 \\ -1-4 & 3+4 \\ 15-0 & 5-20 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} -2 & 10 \\ -5 & 7 \\ 15 & -15 \\ \end{matrix} \right] \end{align} Thus, the value of $AB-4A$ is $\left[ \begin{matrix} -2 & 10 \\ -5 & 7 \\ 15 & -15 \\ \end{matrix} \right]$.