Question

Solve the following equations for $X$ and $Y$
$2X-Y=\left[\begin{array}{ccc}3& -3& 0\\ 3& 3& 2\end{array}\right]$
$2Y+X=\left[\begin{array}{ccc}4& 1& 5\\ -1& 4& -4\end{array}\right]$

Answer 1

Given
$2X-Y=\left[\begin{array}{ccc}3& -3& 0\\ 3& 3& 2\end{array}\right]$
Multiplying both sides by $2$ then
$4X-2Y=2\left[\begin{array}{ccc}3& -3& 0\\ 3& 3& 2\end{array}\right]$
$=\left[\begin{array}{ccc}6& -6& 0\\ 6& 6& 4\end{array}\right]\left(i\right)$
Also, given
$X+2Y=\left[\begin{array}{ccc}4& 1& 5\\ -1& 4& -4\end{array}\right]\left(ii\right)$
Adding $\left(i\right)$ and $\left(ii\right)$, we get
$5X=\left[\begin{array}{ccc}6& -6& 0\\ 6& 6& 4\end{array}\right]+\left[\begin{array}{ccc}4& 1& 5\\ -1& 4& -4\end{array}\right]$
$=\left[\begin{array}{ccc}6+4& -6+1& 0+5\\ 6-1& 6+4& 4-4\end{array}\right]$
$=\left[\begin{array}{ccc}10& -5& 5\\ 5& 10& 0\end{array}\right]$
$\therefore X=\frac{1}{5}\left[\begin{array}{ccc}10& -5& 5\\ 5& 10& 0\end{array}\right]$
$X=\left[\begin{array}{ccc}2& -1& 1\\ 1& 2& 0\end{array}\right]$
Putting the value of $X$ in $\left(ii\right)$, we have
$\left[\begin{array}{ccc}2& -1& 1\\ 1& 2& 0\end{array}\right]+2Y=\left[\begin{array}{ccc}4& 1& 5\\ -1& 4& -4\end{array}\right]$
$\Rightarrow 2Y=\left[\begin{array}{ccc}4& 1& 5\\ -1& 4& -4\end{array}\right]-\left[\begin{array}{ccc}2& -1& 1\\ 1& 2& 0\end{array}\right]$
$=\left[\begin{array}{ccc}4-2& 1+1& 5-1\\ -1-1& 4-2& -4-0\end{array}\right]$
$\therefore Y=\left[\begin{array}{ccc}1& 1& 2\\ -1& 1& -2\end{array}\right]$