College Algebra, Chapter 7, Review Exercises, Section Review Exercises, Problem 42

Solve the matrix equation $AX = B$ 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}

AX =& B
&& \text{Given equation}
\\
\\
X =& \frac{B}{A}
&& \text{Divide by matrix } A
\\
\\
X =& A^{-1} B
&&
\\
\\
X =& \frac{1}{(2)(2) - (1)(3)} \left( \left[ \begin{array}{cc}
2 & -1 \\
-3 & 2
\end{array} \right] \left[ \begin{array}{cc}
1 & -2 \\
-2 & 4
\end{array} \right] \right)
&&
\\
\\
X =& \left[ \begin{array}{cc}
2 & -1 \\
-3 & 2
\end{array} \right]
\left[ \begin{array}{cc}
1 & -2 \\
-2 & 4
\end{array} \right]
&&
\\
\\
X =& \left[ \begin{array}{cc}
2 \cdot 1 + (-1) \cdot (-2) & 2 \cdot (-2) + (-1) \cdot 4 \\
-3 \cdot 1 + 2 \cdot (-2) & -3 \cdot (-2) + 2 \cdot 4
\end{array} \right]
&&
\\
\\
X =& \left[ \begin{array}{cc}
4 & -8 \\
-7 & 14
\end{array} \right]
&&

\end{aligned}
\end{equation}
$