Since $A\in M_{m\times n}(F)$ is an $m\times n$ matrix of rank $k$. Can $A$ be expressed as $A=BC$ where $B$ is an $m\times k$ matrix of rank $k$, and $C$ is a $k\times n$ matrix of rank $k$?
I know that since $A$ has rank $k$, then $A=PJ_kQ$ where $ J_k=\begin{bmatrix} I_k & 0_{k,n-k}\\ 0_{m-k,k} & 0_{m-k,n-k} \end{bmatrix} $ and $P$ is an invertible $m\times m$ matrix, and $Q$ is an invertible $n\times n$ matrix. However, I can't seem to fiddle this into two matrices like $B$ and $C$. Thanks.