Let $A\in \mathbb{M}_{m\times n}(\mathbb{R})$ be of rank $m$. Choose the correct statement(s) from below:
a)The map $\mathbb{R}^n\to \mathbb{R}^m$ given by $v\to Av$ is injective .
b)There exists matrices $B\in \mathbb{M}_m(\mathbb{R})$ and $C\in \mathbb{M}_m(\mathbb{R})$ such that $BAC=[I_m | \mathbb{O}_{n-m}]$
c)There exists matrices $B\in \mathbb{GL}_m(\mathbb{R})$ and $C\in \mathbb{GL}_m(\mathbb{R})$ such that $BAC=[I_m | \mathbb{O}_{n-m}]$
d)For every $(B,C)\in \mathbb{M}_{m}(\mathbb{R}) \times \mathbb{M}_n(\mathbb{R})$ such that $BAC=[I_n| \mathbb{O}_{n-m}]$, $C$ is uniquely determine by $B$
(a) is false as if $m\neq n$ then it cannot be injective.But what we say about else ??For existence of such $B$ matrices how i find such $C$ whcih satisfy it ?? Any help plz ..