## 每週問題 June 5, 2017

Let $A$ and $B$ be $n\times n$ matrices, where $n$ is an odd number. Prove that if $AB=0$ then at least one of the matrices $A+A^T$ and $B+B^T$ is singular.

$\hbox{rank}A+\hbox{rank}B\le\hbox{rank} (AB)+2m+1=2m+1$

$\hbox{rank}(A+A^T)\le \hbox{rank}A+\hbox{rank}A^T\le 2m<2m+1$

This entry was posted in pow 向量空間, 每週問題 and tagged , . Bookmark the permalink.