Let $A \in \mathbb{R}^{n\times n}$ be an invertible matrix with positive entries, and $b \in \mathbb{R}^n$ a vector with positive entries.

When does $A^{-1}b$ have all positive entries? I am looking for references to sufficient (or, ideally, equivalent) conditions.

So far, I found only one by M. Kaykobad (1985): If $$ b_i > \sum_{j\neq i} A_{ij} \frac{b_j}{A_{jj}} $$ for all $i=1,\dots,n,$ then $A^{-1}b$ has all positive entries.

Maybe we can use the fact that $A^{-1}$ is an $M$-matrix, or there are some results from probability theory?