Question

Solve the following system of linear equations, using matrix method

$5x-2y=3,3x+2y=5$

Answer 1

The given system can be written as AX=B, where
$A=\left[\begin{array}{cc}5& 2\\ 3& 2\end{array}\right],X=\left[\begin{array}{c}x\\ y\end{array}\right]andB=\left[\begin{array}{c}3\\ 5\end{array}\right]Here,|A|=\left[\begin{array}{cc}5& 2\\ 3& 2\end{array}\right]=10-6=4\ne 0$,
Thus, A is non-singular, Therefore, its inverse exists.
Therefore, the given system is consistent and has a unique solution given by $X=A^{-1}B$.
Cofactors of A are,
$A_{11}=2,A_{12}=-3,A_{21}=-2,A_{22}=5adj\left(A\right)={\left[\begin{array}{cc}2& -3\\ -2& 5\end{array}\right]}^T=\left[\begin{array}{cc}2& -2\\ -3& 5\end{array}\right]\therefore A^{-1}=\frac{1}{|A|}\left(adjA\right)=\frac{1}{4}\left[\begin{array}{cc}2& -2\\ -3& 5\end{array}\right]$
Now, $X=A^{-1}B=\frac{1}{4}\left[\begin{array}{cc}2& -2\\ -3& 5\end{array}\right]\left[\begin{array}{c}3\\ 5\end{array}\right]=\frac{1}{4}\left[\begin{array}{cc}6& -10\\ -9& +25\end{array}\right]=\frac{1}{4}\left[\begin{array}{c}-4\\ 16\end{array}\right]=\left[\begin{array}{c}-1\\ 4\end{array}\right]$
$\Rightarrow \left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}-1\\ 4\end{array}\right]$, Hence, x =-1 and y=4