College Algebra, Chapter 7, Review Exercises, Section Review Exercises, Problem 40

Solve the matrix equation $2X + C = 5A$ for the unknown matrix, $X$ or show that no solution exists, where

$\displaystyle A = \left[ \begin{array}{cc}
2 & 1 \\
3 & 2
\end{array} \right] \qquad B = \left[ \begin{array}{cc}
1 & -2 \\
-2 & 4
\end{array} \right] \qquad C = \left[ \begin{array}{ccc}
0 & 1 & 3 \\
-2 & 4 & 0
\end{array} \right]$


$
\begin{equation}
\begin{aligned}

2X + C =& 5A
&& \text{Given equation}
\\
\\
2X =& 5A - C
&& \text{Subtract matrix } C
\\
\\
X =& \frac{1}{2} (5A - C)
&& \text{Multiply each side by scalar } \frac{1}{2}
\\
\\
X =& \frac{1}{2} \left( 5\left[ \begin{array}{cc}
2 & 1 \\
3 & 2
\end{array} \right] - \left[ \begin{array}{ccc}
0 & 1 & 3 \\
-2 & 4 & 0
\end{array} \right] \right)
&&
\\
\\
X =& \frac{1}{2} \left( \left[ \begin{array}{cc}
10 & 5 \\
15 & 10
\end{array} \right] - \left[ \begin{array}{ccc}
0 & 1 & 3 \\
-2 & 4 & 0
\end{array} \right] \right)
&&

\end{aligned}
\end{equation}
$


The given equation has no solution because we can't subtract matrices of different dimensions.