Find
Question:

If $X-Y=\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]$ and $X+Y=\left[\begin{array}{rrr}3 & 5 & 1 \\ -1 & 1 & 4 \\ 11 & 8 & 0\end{array}\right]$, find $X$ and $Y$

Solution:

Here,

$X-Y+X+Y=\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]+\left[\begin{array}{ccc}3 & 5 & 1 \\ -1 & 1 & 4 \\ 11 & 8 & 0\end{array}\right]$

$\Rightarrow 2 X=\left[\begin{array}{lll}1+3 & 1+5 & 1+1 \\ 1-1 & 1+1 & 0+4 \\ 1+11 & 0+8 & 0+0\end{array}\right]$

$\Rightarrow 2 X=\left[\begin{array}{lll}4 & 6 & 2 \\ 0 & 2 & 4 \\ 12 & 8 & 0\end{array}\right]$

$\Rightarrow X=\frac{1}{2}\left[\begin{array}{ccc}4 & 6 & 2 \\ 0 & 2 & 4 \\ 12 & 8 & 0\end{array}\right]$

$\Rightarrow X=\left[\begin{array}{lll}2 & 3 & 1 \\ 0 & 1 & 2 \\ 6 & 4 & 0\end{array}\right]$

Now,

$(X-Y)-(X+Y)=\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]-\left[\begin{array}{ccc}3 & 5 & 1 \\ -1 & 1 & 4 \\ 11 & 8 & 0\end{array}\right]$

$\Rightarrow X-Y-X-Y=\left[\begin{array}{ccc}1-3 & 1-5 & 1-1 \\ 1+1 & 1-1 & 0-4 \\ 1-11 & 0-8 & 0-0\end{array}\right]$

$\Rightarrow-2 Y=\left[\begin{array}{ccc}-2 & -4 & 0 \\ 2 & 0 & -4 \\ -10 & -8 & 0\end{array}\right]$

$\Rightarrow Y=-\frac{1}{2}\left[\begin{array}{ccc}-2 & -4 & 0 \\ 2 & 0 & -4 \\ -10 & -8 & 0\end{array}\right]$

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

$\therefore X=\left[\begin{array}{lll}2 & 3 & 1 \\ 0 & 1 & 2 \\ 6 & 4 & 0\end{array}\right]$ and $Y=\left[\begin{array}{ccc}1 & 2 & 0 \\ -1 & 0 & 2 \\ 5 & 4 & 0\end{array}\right]$