College Algebra, Chapter 7, 7.2, Section 7.2, Problem 20

Solve the matrix equation $2A = B - 3X$ for the unknown matrix $X$, where

We solve for $X$


$\begin{equation} \begin{aligned} 2A =& B - 3X && \text{Given equation} \\ \\ 3X =& B - 2A && \text{Add the matrix $3X - 2A$ to each side} \\ \\ X =& \frac{1}{3} (B - 2A) && \text{Multiply each side by the scalar } \frac{1}{3} \end{aligned} \end{equation}$


So,


$\begin{equation} \begin{aligned} X =& \frac{1}{3} \left( \left[ \begin{array}{ccc} 3 & \displaystyle \frac{1}{2} & 5 \\ 1 & -1 & 3 \end{array} \right] - 2 \left[ \begin{array}{cc} 2 & -5 \\ 0 & 7 \end{array} \right] \right) && \text{Substitute the matrices $B$ and $A$} \end{aligned} \end{equation}$


But $B - 2A$ is undefined because we can't add matrices of different dimensions.