College Algebra, Chapter 7, 7.2, Section 7.2, Problem 16

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



$
\begin{equation}
\begin{aligned}

A =& \left[ \begin{array}{cc}
4 & 6 \\
1 & 3
\end{array} \right]

&& B = \left[ \begin{array}{cc}
2 & 5 \\
3 & 7
\end{array} \right]
\\
\\
\\
\\
C =& \left[ \begin{array}{cc}
2 & 3 \\
1 & 0 \\
0 & 2
\end{array} \right]

&& D = \left[ \begin{array}{cc}
10 & 20 \\
30 & 20 \\
10 & 0
\end{array} \right]

\end{aligned}
\end{equation}
$


We solve for $X$


$
\begin{equation}
\begin{aligned}

3X - B =& C
&& \text{Given equation}
\\
\\
3X =& C + B
&& \text{Add the matrix $B$ to each side}
\\
\\
X =& \frac{1}{3} (C + B)
&& \text{Multiply each side b y the scalar } \frac{1}{3}

\end{aligned}
\end{equation}
$


So,

$\displaystyle X = \frac{1}{3} \left( \left[ \begin{array}{cc}
2 & 3 \\
1 & 0 \\
0 & 2
\end{array} \right] + \left[ \begin{array}{cc}
2 & 5 \\
3 & 7
\end{array} \right] \right) \qquad$ Substitute the matrices $C$ and $B$.

But $C + B$ is undefined because we can't add matrices of different dimensions.