Inverse of a stochastic matrix: when is there a strictly negative element?

Question

Let the square matrix $A$ be invertible and left-stochastic (each column adds up to one).

Under which conditions is at least one element of the inverse matrix $A^{-1}$ strictly negative?

For example, if A is bistochastic and orthogonal, then all entries will be non-negative. I'm playing around with examples, but I don't see a pattern. Any suggestions?