Question

Solve system of linear equations , using matrix method.
$2x-y=-2$
$3x+4y=3$

Answer 1

The given system of equations can be written in the form of $AX=B$, where

$A=\left[\begin{array}{cc}2& -1\\ 3& 4\end{array}\right].X=\left[\begin{array}{c}x\\ y\end{array}\right]\text{and}B=\left[\begin{array}{c}-2\\ 3\end{array}\right]$

Now, $\left|A\right|=8+3=11\ne 0$

Thus, $A$ is non-singular. Therefore, its inverse exists.

Now, $A^{-1}=\frac{1}{\left|A\right|}\text{adj}A=\frac{1}{11}\left[\begin{array}{cc}4& 1\\ -3& 2\end{array}\right]$

$\therefore X=A^{-1}B=\frac{1}{11}\left[\begin{array}{cc}4& 1\\ -3& 2\end{array}\right]\left[\begin{array}{c}-2\\ 3\end{array}\right]$

$\Rightarrow \left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{11}\left[\begin{array}{c}-8+3\\ 6+6\end{array}\right]=\frac{1}{11}\left[\begin{array}{c}-5\\ 12\end{array}\right]=\left[\begin{array}{c}\frac{-5}{11}\\ \frac{12}{11}\end{array}\right]$

Hence, $x=\frac{-5}{11}\text{and}y=\frac{12}{11}$.