Counter-example Linear Algebra Question - Rank

Question

Let $B$ be an $n \times m$ matrix, and let $A$ be the $(n-1) \times m$ matrix resulting from removing the last row of $B$.

Is it true that $\text{rank}(A)+1 \geq \text{rank} (B) \geq \text{rank} (A)$?

Also, if it is false, can someone please go through your process in coming up with a counter-example?

Answer 1

HINT If you removed a linearly dependent row (so you did not reduce the rank), then $\mathrm{rank}(B) = \mathrm{rank}(A)$, but if you reduced the rank then $\mathrm{rank}(B) = \mathrm{rank}(A) + 1$.