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 $3(X + Y)$.
Answer & explanation
Correct answer: option 1
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 $A^{-1}=\frac{1}{3}\begin{bmatrix}-1&1\\-4&1\end{bmatrix}$
$⇒X=BA^{-1}=\frac{1}{3}\begin{bmatrix}1&-1\\2&-2\end{bmatrix}\begin{bmatrix}-1&1\\-4&1\end{bmatrix}=\begin{bmatrix}1&0\\2&0\end{bmatrix}$
$⇒Y=A^{-1}B=\frac{1}{3}\begin{bmatrix}-1&1\\-4&1\end{bmatrix}\begin{bmatrix}1&-1\\2&-2\end{bmatrix}=\frac{1}{3}\begin{bmatrix}1&-1\\-2&2\end{bmatrix}$
$⇒3(X+Y)=\begin{bmatrix}3&0\\6&0\end{bmatrix}+\begin{bmatrix}1&-1\\-2&2\end{bmatrix}=\begin{bmatrix}4&-1\\4&2\end{bmatrix}$