Exercise 9, § 10

Let $X, Y$ be two matrices with $m$ rows and $n$ columns over a field $K$; if there exist two square matrices $P, P_1$ of order $m$ and two square matrices $Q, Q_1$ of order $n$ such that $Y = PXQ$ and $X = P_1 X Q_1$, show that $X$ and $Y$ are equivalent.