We know that if a matrix $\operatorname{rank}(A) \leq r$ where $A \in \mathbb{R}^{N \times M}$ then each submatrix $B_i \in \mathbb{R}^{n \times m}$ has $\operatorname{rank}(B_i)\leq r$
The opposite is true also if we take all $B_i \in \mathbb{R}^{n \times m}$ where $N \geq n \geq r$ , $M \geq m \geq r$ and make sure all $\operatorname{rank}(B_i)\leq r$ then $\operatorname{rank}(A) \leq r$
My question is if we take a particular $m,n$ and fix them, say $m=r+1$, $n=r+1$, can we claim if all $\operatorname{rank}(B_i) \leq r$ then $\operatorname{rank}(A) \leq r$.
Thanks a lot.