Question

Find matrices $X$ and $Y$, if $2X-Y=\left[\begin{array}{ccc}6& -6& 0\\ -4& 2& 1\end{array}\right]$ and $X+2Y=\left[\begin{array}{ccc}3& 2& 5\\ -2& 1& -7\end{array}\right]$.

Answer 1

$\Rightarrow$ $2X-Y=\left[\begin{array}{ccc}6& -6& 0\\ -4& 2& 1\end{array}\right]$ ---- ( 1 )

$\Rightarrow$ $X+2Y=\left[\begin{array}{ccc}3& 2& 5\\ -2& 1& -7\end{array}\right]$ ---- ( 2 )

Multipleyng equation ( 1 ) by $2$, we get

$\Rightarrow$ $4X-2Y=\left[\begin{array}{ccc}12& -12& 0\\ -8& 4& 2\end{array}\right]$ ---- ( 3 )

Now adding equation ( 2 ) and ( 3 ) we get,

$\Rightarrow$ $X+2Y+4X-2Y=\left[\begin{array}{ccc}3& 2& 5\\ -2& 1& -7\end{array}\right]+\left[\begin{array}{ccc}12& -12& 0\\ -8& 4& 2\end{array}\right]$

$\Rightarrow$ $5X=\left[\begin{array}{ccc}3+12& 2+(-12)& 5+0\\ -2+(-8)& 1+4& -7+2\end{array}\right]$

$\Rightarrow$ $X=\frac{1}{5}\left[\begin{array}{ccc}15& -10& 5\\ -10& 5& -5\end{array}\right]$

$\Rightarrow$ $X=\left[\begin{array}{ccc}3& -2& 1\\ -2& 1& -1\end{array}\right]$

From ( 2 ),

$\Rightarrow$ $X+2Y=\left[\begin{array}{ccc}3& 2& 5\\ -2& 1& -7\end{array}\right]$

$\Rightarrow$ $2Y=\left[\begin{array}{ccc}3& 2& 5\\ -2& 1& -7\end{array}\right]-X$

Substituting matrix $X,$

$\Rightarrow$ $2Y=\left[\begin{array}{ccc}3& 2& 5\\ -2& 1& -7\end{array}\right]-\left[\begin{array}{ccc}3& -2& 1\\ -2& 1& -1\end{array}\right]$

$\Rightarrow$ $2Y=\left[\begin{array}{ccc}0& 4& 4\\ 0& 0& -6\end{array}\right]$

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

$\therefore$ $Y=\left[\begin{array}{ccc}0& 2& 2\\ 0& 0& -3\end{array}\right]$