Let $A$ and $B$ be two $ m \times n$ matrices , such that $B = PA$ where $P$ is a product of elementary matrices. Now, when $rank(A) < m$ does it necessarily mean that $P$ is not unique and when $rank(A)=m$ does it imply that $P$ is unique?
Verification for special cases:
Suppose, $B=P_{1}A$ and $B=P_{2}A$
$ => 0= (P_{1}-P_{2})A$.
Thus, when $A=0$ i.e. $rank(A)=0<m$, $P$ is not unique. This special case supports my first statement.
Similarly, when $A$ is a square invertible matrix (and thus, so is $B$) then, $rank(A)=m$, and $P=BA^{-1}$ . Thus, $P$ is unique. This scenario supports my second statement.
But, the proof (or counter example) eludes me in both the cases.