Sign of the diagonal elements of the inverse of a matrix

Question

I was wondering if there is any elegant answer to the following question:

Assume that $A$ is Hurwitz (all the eigenvalues of $A$ are on the open left-half complex plane) and has zero or negative diagonal elements.

Is it correct to say that $A^{-1}$ has all zero or negative diagonal elements? Or, do I need any other condition on $A$ for that to be true (if any)?

Thanks in advance!