Question

Find $X$ and $Y$, if $X+Y=\left[\begin{array}{cc}5& 2\\ 0& 9\end{array}\right]$ and $X-Y=\left[\begin{array}{cc}3& 6\\ 0& -1\end{array}\right]$

Answer 1

Given :
$X+Y=\left[\begin{array}{cc}5& 2\\ 0& 9\end{array}\right]\cdots \left(1\right)$
$X-Y=\left[\begin{array}{cc}3& 6\\ 0& -1\end{array}\right]\cdots \left(2\right)$
Adding equation $\left(1\right)$ and $\left(2\right)$
$\Rightarrow (X+Y)+(X-Y)=\left[\begin{array}{cc}5& 2\\ 0& 9\end{array}\right]+\left[\begin{array}{cc}3& 6\\ 0& -1\end{array}\right]$
$\Rightarrow X+X+Y-Y=\left[\begin{array}{cc}5+3& 2+6\\ 0+0& 9-1\end{array}\right]$
$\Rightarrow 2X=\left[\begin{array}{cc}8& 8\\ 0& 8\end{array}\right]$
$\Rightarrow X=\frac{1}{2}\left[\begin{array}{cc}8& 8\\ 0& 8\end{array}\right]$
$\Rightarrow X=\left[\begin{array}{cc}\frac{8}{2}& \frac{8}{2}\\ 0& \frac{8}{2}\end{array}\right]$
$\Rightarrow X=\left[\begin{array}{cc}4& 4\\ 0& 4\end{array}\right]$

Putting $X=\left[\begin{array}{cc}4& 4\\ 0& 4\end{array}\right]$ in equation $\left(1\right)$
$\Rightarrow \left[\begin{array}{cc}4& 4\\ 0& 4\end{array}\right]+Y=\left[\begin{array}{cc}5& 2\\ 0& 9\end{array}\right]$

$\Rightarrow Y=\left[\begin{array}{cc}5& 2\\ 0& 9\end{array}\right]-\left[\begin{array}{cc}4& 4\\ 0& 4\end{array}\right]$

$\Rightarrow Y=\left[\begin{array}{cc}5-4& 2-4\\ 0-0& 9-4\end{array}\right]$

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