Solution of simultaneous equations using matrices
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Solve simultaneously \(\begin{array}{*{20}{c}}{2x + 3y = 13}\\{3x + 2y = 17}\end{array}\)
\begin{align}
&\begin{aligned}
\text{Let } A&=\left[\begin{array}{ll}2 & 3 \\ 3 & 2\end{array}\right] \text{ and } C=\left[\begin{array}{l}13 \\ 17\end{array}\right]\\
A^{-1}&=\frac{1}{(4-9)}\left[\begin{array}{cc}
2 & -3 \\
-3 & 2
\end{array}\right]
\end{aligned}\\
&\begin{aligned}
B &=A^{-1} C \\
&=-\frac{1}{5}\left[\begin{array}{cc}
2 & -3 \\
-3 & 2
\end{array}\right]\left[\begin{array}{l}
13 \\
17
\end{array}\right] \\
&=-\frac{1}{5}\left[\begin{array}{l}
26-51 \\
-39+34
\end{array}\right] \\
&=-\frac{1}{5}\left[\begin{array}{l}
-25 \\
-5
\end{array}\right]=\left[\begin{array}{l}
5 \\
1
\end{array}\right] \\
&\quad \therefore x=5,\ y=1
\end{aligned}
\end{align}
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