## Precalculus (6th Edition) Blitzer

The matrix $X$ is, $X=\left[ \begin{matrix} -2 & -6 \\ 3 & \frac{1}{3} \\ \end{matrix} \right]$
Consider the given matrix equation $3X+A=B$. Now put the matrices $A$ and $B$ in the above matrix equation and then simplify for $X$ as below: $3X+\left[ \begin{matrix} 4 & 6 \\ -5 & 0 \\ \end{matrix} \right]=\left[ \begin{matrix} -2 & -12 \\ 4 & 1 \\ \end{matrix} \right]$ Now subtract the matrix $\left[ \begin{matrix} 4 & 6 \\ -5 & 0 \\ \end{matrix} \right]$ from both sides of the above expression to get, \begin{align} & 3X=\left[ \begin{matrix} -2 & -12 \\ 4 & 1 \\ \end{matrix} \right]-\left[ \begin{matrix} 4 & 6 \\ -5 & 0 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} -2-4 & -12-6 \\ 4+5 & 1-0 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} -6 & -18 \\ 9 & 1 \\ \end{matrix} \right] \end{align} So, $3X=\left[ \begin{matrix} -6 & -18 \\ 9 & 1 \\ \end{matrix} \right]$ Dividing both sides by 3 we get, \begin{align} & X=\frac{1}{3}\left[ \begin{matrix} -6 & -18 \\ 9 & 1 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} -2 & -6 \\ 3 & \frac{1}{3} \\ \end{matrix} \right] \end{align} Thus, $X=\left[ \begin{matrix} -2 & -6 \\ 3 & \frac{1}{3} \\ \end{matrix} \right]$