Given matrices $A$ and $B$, solve $XA = B$

Question

Let $$A = \begin{bmatrix} 3&-7\\ 1&-2\end{bmatrix} \qquad \qquad B = \begin{bmatrix} 0&3\\ 1&-5\end{bmatrix}$$ and $X$ be an unknown $2x2$ matrix.

a. Find $A^{-1}$

b. If $XA = B$, use (a) to find $X$.

I found

$$A^{-1} = \begin{bmatrix} -2&7\\ -1&3\end{bmatrix}$$

I am stuck on the part b. I thought that if $XA=B$, then

$$X=A^{-1}B$$

so I did:

$$ X= \left[ \begin{array}{cc} -2&7\\ -1&-3 \end{array} \right] \left[ \begin{array}{cc} 0&3\\ 1&-5 \end{array} \right] $$

and got:

$$X = \left[ \begin{array}{cc} 7&-41\\ 3&-18 \end{array} \right] $$

I have been told that this is not correct and I missed a technical detail of matrix multiplication. Please help.

Answer 1

Recall that we need to make a right multiplication by $A^{-1}$

$$XA=B\implies XAA^{-1}=BA^{-1}\implies X=BA^{-1}$$

Answer 2

The problem with your answer lies in the fact that what we actually have is$$XA=B\iff X=BA^{-1}.$$