Let $A$ be a non-negative invertible integer matrix. Furthermore, let $P$ be any row-stochastic matrix.
For which $A$ exists $P$ such that $PA \succ 0$?
Results regarding the restriction of $P$ to permutations would already be helpful. For example, $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is indefinite while exchanging its rows renders it positive definite. What can be said in general? Does anything change when one drops the integer constraint, or imposes symmetry on $A$?