Question

Find $X$ and $Y$, if
$\left(i\right)$ $X+Y=\left[\begin{array}{cc}7& 0\\ 2& 5\end{array}\right]$ and $X-Y=\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]$
$\left(ii\right)$ $2X+3Y=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]$ and $3X+2Y=\left[\begin{array}{cc}2& -2\\ -1& 5\end{array}\right]$

Answer 1

$\left(i\right)$ Given : $X+Y=\left[\begin{array}{cc}7& 0\\ 2& 5\end{array}\right]\cdots \left(i\right)$
and $X-Y=\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]\cdots \left(ii\right)$
Adding equation $\left(i\right)$ and $\left(ii\right)$
$X+Y+X-Y=\left[\begin{array}{cc}7& 0\\ 2& 5\end{array}\right]+\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]$
$X+Y++X-Y=\left[\begin{array}{cc}7+3& 0+0\\ 2+0& 5+3\end{array}\right]$
$\Rightarrow 2X=\left[\begin{array}{cc}10& 0\\ 2& 8\end{array}\right]$
$\Rightarrow X=\frac{1}{2}\left[\begin{array}{cc}10& 0\\ 2& 8\end{array}\right]$
$\Rightarrow X=\left[\begin{array}{cc}\frac{10}{2}& \frac{0}{2}\\ \frac{2}{2}& \frac{8}{2}\end{array}\right]=\left[\begin{array}{cc}5& 0\\ 1& 4\end{array}\right]$
Putting value of $X$ in equation $\left(i\right)$
$\Rightarrow X+Y=\left[\begin{array}{cc}7& 0\\ 2& 5\end{array}\right]$
$\Rightarrow Y=\left[\begin{array}{cc}7& 0\\ 2& 5\end{array}\right]-X$
$\Rightarrow Y=\left[\begin{array}{cc}7& 0\\ 2& 5\end{array}\right]-\left[\begin{array}{cc}5& 0\\ 1& 4\end{array}\right]$
$\Rightarrow Y=\left[\begin{array}{cc}7-5& 0-0\\ 2-1& 5-4\end{array}\right]$
$\Rightarrow Y=\left[\begin{array}{cc}2& 0\\ 1& 1\end{array}\right]$
Hence, $X=\left[\begin{array}{cc}5& 0\\ 1& 4\end{array}\right]$ and $Y=\left[\begin{array}{cc}2& 0\\ 1& 1\end{array}\right]$


$\left(ii\right)$ Given: $2X+3Y=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]\cdots \left(i\right)$
and $3X+2Y=\left[\begin{array}{cc}2& -2\\ -1& 5\end{array}\right]\cdots \left(ii\right)$
Multiplying equation $\left(i\right)$ by $3$
$\Rightarrow 3\times (2X+3Y)=3\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]$
$\Rightarrow 6X+9Y=\left[\begin{array}{cc}2\times 3& 3\times 3\\ 4\times 3& 0\times 3\end{array}\right]$
$\Rightarrow 6X+9Y=\left[\begin{array}{cc}6& 9\\ 12& 0\end{array}\right]\cdots \left(iii\right)$
$3X+2Y=\left[\begin{array}{cc}2& -2\\ -1& 5\end{array}\right]\cdots \left(ii\right)$
Multiplying equation $\left(ii\right)$ by $2$
$\Rightarrow 2\times (3X+2Y)=2\times \left[\begin{array}{cc}2& -2\\ -1& 5\end{array}\right]$
$\Rightarrow 6X+4Y=\left[\begin{array}{cc}2\times 2& -2\times 2\\ -1\times 2& 5\times 2\end{array}\right]$
$\Rightarrow 6X+4Y=\left[\begin{array}{cc}4& -4\\ -2& 10\end{array}\right]\cdots \left(iv\right)$

Solve for value of $X$ and $Y$
Subtracting $\left(iv\right)$ from $\left(iii\right)$
$\Rightarrow (6X+9Y-(6X+4Y)=\left[\begin{array}{cc}6& 9\\ 12& 0\end{array}\right]-\left[\begin{array}{cc}4& -4\\ -2& 10\end{array}\right]$
$\Rightarrow 6X+9Y-6X-4Y=\left[\begin{array}{cc}6-4& 9-(-4)\\ 12-\left(2\right)& 0-10\end{array}\right]$
$\Rightarrow 9Y-4Y+6X-6X=\left[\begin{array}{cc}2& 9+4\\ 12+2& -10\end{array}\right]$
$\Rightarrow 5Y+0=\left[\begin{array}{cc}2& 13\\ 14& -10\end{array}\right]$
$\Rightarrow Y=\frac{1}{5}\left[\begin{array}{cc}2& 13\\ 14& -10\end{array}\right]$
$\Rightarrow Y=\left[\begin{array}{cc}\frac{2}{5}& \frac{13}{5}\\ \frac{14}{5}& -\frac{10}{5}\end{array}\right]$
$\Rightarrow Y=\left[\begin{array}{cc}\frac{2}{5}& \frac{13}{5}\\ \frac{14}{5}& -2\end{array}\right]$
Putting value of $Y$ in equation $\left(i\right)$
$2X+3Y=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]$
$\Rightarrow 2X+3\left[\begin{array}{cc}\frac{2}{5}& \frac{13}{5}\\ \frac{14}{5}& -2\end{array}\right]=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]$
$\Rightarrow 2X+\left[\begin{array}{cc}3\times \frac{2}{5}& 3\times \frac{13}{5}\\ 3\times \frac{14}{5}& 3\times -2\end{array}\right]=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]$
$\Rightarrow 2X+\left[\begin{array}{cc}\frac{6}{5}& \frac{39}{5}\\ \frac{42}{5}& -6\end{array}\right]=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]$
$\Rightarrow 2X=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]-\left[\begin{array}{cc}\frac{6}{5}& \frac{39}{5}\\ \frac{42}{5}& -6\end{array}\right]$
$\Rightarrow 2X=\left[\begin{array}{cc}2-\frac{6}{5}& 3-\frac{39}{5}\\ 4-\frac{42}{5}& 0-(-6)\end{array}\right]$
$\Rightarrow 2X=\left[\begin{array}{cc}\frac{4}{5}& \frac{-24}{5}\\ \frac{-22}{5}& 6\end{array}\right]$
$\Rightarrow X=\frac{1}{2}\left[\begin{array}{cc}\frac{4}{5}& \frac{-24}{5}\\ \frac{-22}{5}& 6\end{array}\right]$
$\Rightarrow X=\left[\begin{array}{cc}\frac{4}{5}\times \frac{1}{2}& \frac{-24}{5}\times \frac{1}{2}\\ \frac{-22}{5}\times \frac{1}{2}& 6\times \frac{1}{2}\end{array}\right]$
$\Rightarrow X=\left[\begin{array}{cc}\frac{2}{5}& \frac{-12}{5}\\ \frac{-11}{5}& 3\end{array}\right]$