I know that rank (A+B)<= rank(A)+rank(B) but why is rank(A+B)>=rank(A)-rank(B)?

Question

Why is $\operatorname{rank}(A+B)\ge \operatorname{rank}(A)-\operatorname{rank}(B)$?

It's obviously true if $rank(A)<rank(B)$ since rank(A+B) can't be negative but what if $rank(A)>rank(B)$?

Answer 1

You can deduce it from the first fact: $$\operatorname{rank}(A) \le \operatorname{rank}(A+B) + \operatorname{rank}(-B) = \operatorname{rank}(A+B) + \operatorname{rank}(B) \implies \\ \operatorname{rank}(A) - \operatorname{rank}(B) \le \operatorname{rank}(A+B).$$