Let $L$ be a $p\times r $ matrix with $r < p$ such that $\text{rk}\, L = r$. If I take the last $r$ rows by carrying out the following:
$\left[ 0 \, \vdots \, I_{r} \right ]\, L = L_{r}$ where $L_r$ is now an $r \times r$ matrix, am I correct in concluding that the $\text{rk}\, L_r = r$ so that $L_r$ becomes invertible? If so, how could I prove this?
All help is warmly appreciated!