## 每週問題 May 23, 2016

Let $A$ and $B$ be $m\times n$ matrices. If $\hbox{rank}A=r$ and $\hbox{rank}B=k\le r$, show that

$r-k\le \hbox{rank}(A+B)\le r+k$.

In words, a perturbation of rank $k$ can change the rank by at most $k$.

$\hbox{rank}(A+B)=\hbox{rank}(A-(-B))\ge \hbox{rank}A-\hbox{rank}(-B)=\hbox{rank}A-\hbox{rank}B=r-k$