Matrix equation problem

Question

Is my solution of this matrix equation correct ?

$(AX^{-1}+B)^{-1}=\frac13X$, I've started off by raising the equation to $-1$ and in the end I get : $((\frac13I-A)^{-1}B)^{-1}$

Answer 1

\begin{eqnarray} (AX^{-1}+B)^{-1} & = & \frac{1}{3}X\\ AX^{-1}+B & = & 3X^{-1}\\ B & = & (3I-A)X^{-1}\\ BX & = & (3I-A)\\ X & = & B^{-1}(3I-A) \end{eqnarray}