Question

If $X$ and $Y$ are $2\times 2$ matrices, then solve the following matrix equations for $X$ and $Y$.
$2X+3Y=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right],3X+2Y=\left[\begin{array}{cc}-2& 2\\ 1& -5\end{array}\right]$

Answer 1

$2X+3Y=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]$ ........$\left(1\right)$

$3X+2Y=\left[\begin{array}{cc}-2& 2\\ 1& -5\end{array}\right]$ ........$\left(2\right)$

$3\times \left(1\right)-2\times \left(2\right)$

$\Rightarrow 6X+9Y=\left[\begin{array}{cc}6& 9\\ 12& 0\end{array}\right]$

$\Rightarrow 6X+4Y=\left[\begin{array}{cc}-4& 4\\ 4& -10\end{array}\right]$

$\Rightarrow 6X+9Y-6X-4Y=\left[\begin{array}{cc}6+4& 9-4\\ 12-4& 0+10\end{array}\right]$

$\Rightarrow 5Y=\left[\begin{array}{cc}10& 5\\ 8& 10\end{array}\right]$

$\Rightarrow Y=\left[\begin{array}{cc}2& 1\\ \frac{8}{5}& 2\end{array}\right]$

We have from $\left(1\right)$

$2X+3Y=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]$

$\Rightarrow 2X=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]-3Y$

$\Rightarrow 2X=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]-3\left[\begin{array}{cc}2& 1\\ \frac{8}{5}& 2\end{array}\right]$

$\Rightarrow 2X=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]-\left[\begin{array}{cc}6& 3\\ \frac{24}{5}& 6\end{array}\right]$

$\Rightarrow 2X=\left[\begin{array}{cc}2-6& 3-3\\ 4-\frac{24}{5}& 0-6\end{array}\right]$

$\Rightarrow 2X=\left[\begin{array}{cc}-4& 0\\ \frac{20-24}{5}& -6\end{array}\right]$

$\Rightarrow 2X=\left[\begin{array}{cc}-4& 0\\ \frac{-4}{5}& -6\end{array}\right]$

$\therefore X=\left[\begin{array}{cc}-2& 0\\ \frac{-2}{5}& -3\end{array}\right]$