## 每週問題 May 16, 2016

Let $A$ and $B$ be $m\times n$ matrices. Show that

$|\hbox{rank}A-\hbox{rank}B|\le \hbox{rank}(A-B)$.

$\hbox{rank}A=\hbox{rank}(A-B+B)\le \hbox{rank}(A-B)+\hbox{rank}B$

$\hbox{rank}B=\hbox{rank}(B-A+A)\le \hbox{rank}(B-A)+\hbox{rank}A=\hbox{rank}(A-B)+\hbox{rank}A$