Question
Find matrices $X$ and $Y, $ if $X+Y=\left[\begin{array}{ll}5 & 2 \\ 0 & 9\end{array}\right]$ and $X Y=\left[\begin{array}{cc}3 & 6 \\ 0 & -1\end{array}\right]$
✓
Answer
We have
$ X+Y=\left[\begin{array}{ll} 5 & 2 \\ 0 & 9 \end{array}\right] $
and $ X-Y=\left[\begin{array}{cc} 3 & 6 \\ 0 & -1 \end{array}\right] $
Now $(X+Y)+(X-Y)=\left[\begin{array}{ll}5 & 2 \\ 0 & 9\end{array}\right]\left[\begin{array}{cc}3 & 6 \\ 0 & -1\end{array}\right]$
$ 2 X=\left[\begin{array}{ll} 8 & 8 \\ 0 & 8 \end{array}\right] $
$X=\frac{1}{2}\left[\begin{array}{ll} 8 & 8 \\ 0 & 8 \end{array}\right]=\left[\begin{array}{ll} 4 & 4 \\ 0 & 4
\end{array}\right]$
Also $(X+Y)-(X-Y)=\left[\begin{array}{ll}5 & 2 \\ 0 & 9\end{array}\right]+\left[\begin{array}{cc}-3 & -6 \\ 0 & 1\end{array}\right]$
$\Rightarrow 2 Y=\left[\begin{array}{cc} 2 & -4 \\ 0 & 10 \end{array}\right]$
$ X+Y=\left[\begin{array}{ll} 5 & 2 \\ 0 & 9 \end{array}\right] $
and $ X-Y=\left[\begin{array}{cc} 3 & 6 \\ 0 & -1 \end{array}\right] $
Now $(X+Y)+(X-Y)=\left[\begin{array}{ll}5 & 2 \\ 0 & 9\end{array}\right]\left[\begin{array}{cc}3 & 6 \\ 0 & -1\end{array}\right]$
$ 2 X=\left[\begin{array}{ll} 8 & 8 \\ 0 & 8 \end{array}\right] $
$X=\frac{1}{2}\left[\begin{array}{ll} 8 & 8 \\ 0 & 8 \end{array}\right]=\left[\begin{array}{ll} 4 & 4 \\ 0 & 4
\end{array}\right]$
Also $(X+Y)-(X-Y)=\left[\begin{array}{ll}5 & 2 \\ 0 & 9\end{array}\right]+\left[\begin{array}{cc}-3 & -6 \\ 0 & 1\end{array}\right]$
$\Rightarrow 2 Y=\left[\begin{array}{cc} 2 & -4 \\ 0 & 10 \end{array}\right]$
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