Proposition: Upper and lower rank certificates

If $N=BT$ and $B$ has $r$ columns, this gives $\mathrm{rank}N\le r$. A nonzero $r\times r$ minor of $N$ gives the reverse bound. It suffices that $B$ consist of original columns of $N$ and contain that minor.

Proof. The image of $N$ lies in the image of $B$. A nonzero original minor gives $r$ independent original columns and rows.