# $B^{-1}$ has one positive entry per row if $B$ is invertible and nonnegative

I'm a little stuck into a question of the book "Linear Programming and Network Flows":

Let $$B$$ be an invertible matrix with nonnegative entries. Show that every row of $$B^{-1}$$ has at least one positive entry.

I tried to show that the elements in the diagonal should be positive in $$B^{-1}$$ when doing elementary operations into $$B$$. But I was not able to go very far...

Suppose that this occurs at row $$j$$. Then we must have $$1 = I_{jj} = (B^{-1}B)_{jj} = \sum_{s}B^{-1}_{js}B_{js},$$ which is absurd since the right hand side is negative (recall that by hypotehsis it is $$B_{js}^{-1}B_{js} \leq 0$$ for all $$s$$).
Seeking a contradiction, suppose that $$B^{-1}$$ has a row $$r$$ whose entries are nonpositive. The equation $$B^{-1}B$$ implies that $$r\cdot b=1$$ for one of the columns $$b$$ of $$B$$. Since each entry in $$r$$ is $$\leq 0$$ and each entry in $$b$$ is $$\geq 0$$, it follows that each term in $$r\cdot b$$ is $$\leq 0$$. Thus $$1=r\cdot b\leq 0$$, a contradiction.