When a Matrix and Its Inverse Are Nonnegative

Abstract

In this article we prove that $A$ and $A^{-1}$ are stochastic if and only of $A$ is a permutation matrix. Then we extend this result to show that $A$ and $A^{-1}$ are nonnegative if and only if it is a product of a diagonal matrix with all positive diagonal entries and a permutation matrix.