Prove that

$\displaystyle A$ be $\displaystyle m\times n$ matrix with rank $\displaystyle r$ iff

there is an invertible $\displaystyle m\times m$ matrix $\displaystyle X$ and an invertible $\displaystyle n\times n$ matrix $\displaystyle Y$ such that $\displaystyle XAY=\begin{pmatrix}I_r & 0\\0 & 0 \end{pmatrix}$, where $\displaystyle I_r$ is $\displaystyle r\times r$ identity marix.