Find the sum of all elements in X  where $2\mathrm{X}-\mathrm{Y}=\left[\begin{array}{ccc}3& -3& 0\\ 3& 3& 2\end{array}\right],2\mathrm{Y}+\mathrm{X}=\left[\begin{array}{ccc}4& 1& 5\\ -1& 4& -4\end{array}\right]$

Given,

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

On multiplying both sides by 2, we get

$4\mathrm{X}-2\mathrm{Y}=2\left[\begin{array}{ccc}3& -3& 0\\ 3& 3& 2\end{array}\right];4\mathrm{X}-2\mathrm{Y}=\left[\begin{array}{ccc}6& -6& 0\\ 6& 6& 4\end{array}\right]----\left(i\right)$

also given $\mathrm{X}+2\mathrm{Y}=\left[\begin{array}{r}4\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}5\\ -1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}4\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-4\end{array}\right]----\left(\mathrm{ii}\right)$

Adding Eqs. (i) and (i), we get 

$\begin{array}{l}5\mathrm{X}=\left[\begin{array}{ccc}6& -6& 0\\ 6& 6& 4\end{array}\right]+\left[\begin{array}{rrr}4& 1& 5\\ -1& 4& -4\end{array}\right]\\ =\left[\begin{array}{lll}6+4& -6+1& 0+5\\ 6-1& 6+4& 4-4\end{array}\right]=\left[\begin{array}{ccc}10& -5& 5\\ 5& 10& 0\end{array}\right]\\ \mathrm{X}=\frac{1}{5}\left[\begin{array}{ccc}10& -5& 5\\ 5& 10& 0\end{array}\right]\Rightarrow \mathrm{X}=\left[\begin{array}{ccc}2& -1& 1\\ 1& 2& 0\end{array}\right]\end{array}$

Putting the value of X in Eq. (i), we get

$\begin{array}{l}\left[\begin{array}{rrr}2& -1& 1\\ 1& 2& 0\end{array}\right]+2\mathrm{Y}=\left[\begin{array}{rrr}4& 1& 5\\ -1& 4& -4\end{array}\right]\\ \Rightarrow 2\mathrm{Y}=\left[\begin{array}{lrr}4& 1& 5\\ -1& 4& -4\end{array}\right]-\left[\begin{array}{rrr}2& -1& 1\\ 1& 2& 0\end{array}\right]\\ =\left[\begin{array}{rrr}4-2& 1+1& 5-1\\ -1-1& 4-2& -4-0\end{array}\right]=\left[\begin{array}{rrr}2& 2& 4\\ -2& 2& -4\end{array}\right]\\ \mathrm{Y}=\left[\begin{array}{rrr}1& 1& 2\\ -1& 1& -2\end{array}\right]\\ \therefore \mathrm{X}=\left[\begin{array}{rrr}2& -1& 1\\ 1& 2& 0\end{array}\right]\text{ and }\mathrm{Y}=\left[\begin{array}{rrr}1& 1& 2\\ -1& 1& -2\end{array}\right]\end{array}$