How to prove the following inequality about the rank of sum of two matrices?

Question

If we have two matrices $A,B$ of same dimension then how can we prove that $$Rank[A]-Rank[B]\leq Rank[A+B]?$$. I know that $$Rank[A]+Rank[B]\geq Rank[A+B]$$, but I do not know how to use this to prove $$Rank[A]-Rank[B]\leq Rank[A+B].$$

Answer 1

Using the known inequality, we have $$\operatorname{rank}[A] = \operatorname{rank}[(A+B)+(-B)] \leq \operatorname{rank}[A+B] + \operatorname{rank}[-B]$$

Also, $\operatorname{rank}[-B] = \operatorname{rank}[B]$, so this gives $$\operatorname{rank}[A] \leq \operatorname{rank}[A+B] + \operatorname{rank}[B]$$ Rearrange to find your inequality.