$A^{-1}XB = I$ Solve for X matrix equation

Question

$A^{-1}XB = I$, $A$ and $B$ are given and they are square matrixes.

If I want to solve this matrix equation for $X$, I need to change it to the form like this $X = A×B×I$?

Answer 1

If $B$ is invertible, i.e. $B^{-1}$ exists, then, multiplying by $A$ from the left and by $B^{-1}$ from the right, you have:

\begin{equation} A^{-1}XB = I\iff X=AA^{-1}XBB^{-1} = AIB^{-1}=AB^{-1}, \end{equation}

using that $AI=A$, respectively $IB^{-1}=B^{-1}$.