Matrices equation with three matrices

Question

I have got a matrices equation: $A \times X \times B = C$ where X is the unknown matrix.

Am I allowed to multiply with the inverse of $(A\times B)$ immediately or... what should I do?

Answer 1

Look at the determinant of the given by equation. We get $$\det (A) \det (X) \det (B) = \det (C),$$ so if $C$ is invertible, that it, $\det (C) \neq 0$, then so are the determinants of $A,X$ and $B$. If that's the case you can pre multiply with $A^{-1}$ and postmultiply with $B^{-1}$ to obtain $$X = A^{-1}CB^{-1}.$$