Given the matrices A and B as $A=\begin{bmatrix}1&-1\\4&-1\end{bmatrix}$ and $B=\begin{bmatrix}1&-1\\2&-2\end{bmatrix}$. The two matrices X and Y are such that $XA = B$ and $AY = B$, then find the matrix $3(X + Y)$.

$\begin{bmatrix}4&-1\\4&2\end{bmatrix}$

Here A is non singular but B is singular hence only $A^{-1}$ exists 

Now $XA=B$

or $X= BA^{-1}$   ...(1)

And $AY=B$

or $Y=A^{-1}B$   ...(2)

Also $A^{-1}=\frac{1}{3}\begin{bmatrix}-1&1\\-4&1\end{bmatrix}$

$⇒X=BA^{-1}=\frac{1}{3}\begin{bmatrix}1&-1\\2&-2\end{bmatrix}\begin{bmatrix}-1&1\\-4&1\end{bmatrix}=\begin{bmatrix}1&0\\2&0\end{bmatrix}$

$⇒Y=A^{-1}B=\frac{1}{3}\begin{bmatrix}-1&1\\-4&1\end{bmatrix}\begin{bmatrix}1&-1\\2&-2\end{bmatrix}=\frac{1}{3}\begin{bmatrix}1&-1\\-2&2\end{bmatrix}$

$⇒3(X+Y)=\begin{bmatrix}3&0\\6&0\end{bmatrix}+\begin{bmatrix}1&-1\\-2&2\end{bmatrix}=\begin{bmatrix}4&-1\\4&2\end{bmatrix}$