$\operatorname{rank}(A+B)\geq|\operatorname{rank}(A)-\operatorname{rank}(B)|$

Question

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$\operatorname{rank}(A+B)\geq|\operatorname{rank}(A)-\operatorname{rank}(B)|$

I have recently started learning linear algebra, although I have solved some rank problems and have seen some inequalities involving rank of a matrix but I dont know what is this inequality, I have not seen this before, a quick google search also doesnot help

I would want to know if there exist such inequality and maybe a proof of the same.

Answer 1

Hint: Let us show that rank $A$ $-$ rank $B$ $\leq$ rank($A + B$). We have $$\operatorname{rank} A = \operatorname{rank} [(A+B)+(-B)] \leq \operatorname{rank}(A+B) + \operatorname{rank}(-B),$$ by the inequality in my comment above.