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.