Find matrices X and Y, if X + Y

Question:

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

Solution:

Given : $(X+Y)+(X-Y)=\left[\begin{array}{ll}5 & 2 \\ 0 & 9\end{array}\right]+\left[\begin{array}{cc}3 & 6 \\ 0 & -1\end{array}\right]$

$\Rightarrow 2 X=\left[\begin{array}{ll}5+3 & 2+6 \\ 0+0 & 9-1\end{array}\right]$

$\Rightarrow 2 X=\left[\begin{array}{ll}8 & 8 \\ 0 & 8\end{array}\right]$

$\Rightarrow X=\frac{1}{2}\left[\begin{array}{ll}8 & 8 \\ 0 & 8\end{array}\right]$

$\Rightarrow X=\left[\begin{array}{ll}4 & 4 \\ 0 & 4\end{array}\right]$

Now,

$(X+Y)-(X-Y)=\left[\begin{array}{ll}5 & 2 \\ 0 & 9\end{array}\right]-\left[\begin{array}{cc}3 & 6 \\ 0 & -1\end{array}\right]$

$\Rightarrow X+Y-X+Y=\left[\begin{array}{ll}5-3 & 2-6 \\ 0-0 & 9+1\end{array}\right]$

$\Rightarrow 2 Y=\left[\begin{array}{ll}2 & -4 \\ 0 & 10\end{array}\right]$

$\Rightarrow Y=\frac{1}{2}\left[\begin{array}{cc}2 & -4 \\ 0 & 10\end{array}\right]$

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

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