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.