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

Given matrix equations :

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

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

Adding $\left(i\right)$ and $\left(ii\right)$

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

$\Rightarrow 5X+5Y=\left[\begin{array}{cc}0& 5\\ 5& -5\end{array}\right]$

$\Rightarrow X+Y=\frac{1}{5}\left[\begin{array}{cc}0& 5\\ 5& -5\end{array}\right]$

$\Rightarrow X+Y=\left[\begin{array}{cc}0& 1\\ 1& -1\end{array}\right]$ ...$\left(iii\right)$

On subtracting $\left(i\right)$ from $\left(ii\right)$ we get

$(3X+2Y)-(2X+3Y)=\left[\begin{array}{cc}-2-2& 2-3\\ 1-4& -5-0\end{array}\right]$

$\Rightarrow X-Y=\left[\begin{array}{cc}-4& -1\\ -3& -5\end{array}\right]$ ...$\left(iv\right)$

Simplifying the equations $\left(iii\right)$ and $\left(iv\right)$
On adding eq. $\left(iii\right)$ and $\left(iv\right)$, we get

$(X+Y)+(X-Y)=\left[\begin{array}{cc}0-4& 1-1\\ 1-3& -1-5\end{array}\right]$

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

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

Putting $X$ in equation $\left(iii\right)$

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

$\Rightarrow Y=\left[\begin{array}{cc}0& 1\\ 1& -1\end{array}\right]-\left[\begin{array}{cc}-2& 0\\ -1& -3\end{array}\right]$

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

Hence, $X=\left[\begin{array}{cc}-2& 0\\ -1& -3\end{array}\right]$ and $Y=\left[\begin{array}{cc}2& 1\\ 2& 2\end{array}\right]$