$|\text{rank}A-\text{rank}B|\leq \text{rank}(A+B)$

Question

I would like to prove the following inequalities:

$|\text{rank}A-\text{rank}B|\leq \text{rank}(A+B)$

I know that $\text{rank}(A+B)\leq \text{rank}A + \text{rank} B,$ but I can't tackle the problem.

Any help would be appreciated.

Thanks

Answer 1

Use the inequality you have already.

We have $$\def\rank{\mathop{\rm rank}}\rank(A)=\rank((A+B)+(-B))\le\rank(A+B)+\rank(-B)\ .$$ See if you can finish it from here.