Exercise 10, § 10

Let $X, X', Y, Y'$ be four square matrices of order $n$ over a ring $A$, such that $X$ is invertible. For there to exist two invertible square matrices $P, Q$ of order $n$ such that $X' = PXQ$ and $Y' = PYQ$, it is necessary and sufficient that $X'$ be invertible and that the matrices $YX^{-1}$ and $Y'{X'}^{-1}$ be similar.