## 每週問題 December 26, 2016

Let $A$ be an $n\times n$ real matrix. Show that $A$ and $A^{-1}$ have all elements nonnegative if and only if each row and each column of $A$ has exactly one positive element and the rest of the elements are zeros.

$\displaystyle 0=(AB)_{ij}=\sum_{k=1}^na_{ik}b_{kj}=a_{ip}b_{pj}+a_{iq}b_{qj}$

### One Response to 每週問題 December 26, 2016

1. Lin 說道：

感觉上和permutation matrix的性质好像啊！permutation matrix A满足每一个行每一个列仅有一个元素为1，其余元素都为0。A的逆矩阵等于A的转置矩阵。