Question

Solve $2x-y=7$; $3x-2y=11$ by using matrix inversion method.

Answer 1

Writing the equations in matrix form, we get

$\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}7\\ 11\end{array}\right]$

(i.e.,) $AX=B\Rightarrow X=A^{-1}B$

where $A=\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]$; $X=\left[\begin{array}{c}x\\ y\end{array}\right]$ and $B=\left[\begin{array}{c}7\\ 11\end{array}\right]$

To find $A^{-1}$:

$A=\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]$

$\left[\begin{array}{c}A\end{array}\right]=\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]=-4+3=-1\ne 0$

$\text{adj}A=\left[\begin{array}{cc}-2& 1\\ -3& 2\end{array}\right]$

$A^{-1}=\frac{1}{\left[\begin{array}{c}A\end{array}\right]}\left(\text{adj}A\right)=\frac{1}{-1}\left[\begin{array}{cc}-2& 1\\ -3& 2\end{array}\right]=\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]$

$X=A^{-1}B\Rightarrow \left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]\left[\begin{array}{c}7\\ 11\end{array}\right]$

$=\left[\begin{array}{c}14-11\\ 21-22\end{array}\right]=\left[\begin{array}{c}3\\ -1\end{array}\right]$

$\Rightarrow x=3,y=-1$