Let $A \in \text{GL}(n,\mathbb R)$ with all nonnegative entries such that $A^{-1}$ also has all nonnegative entries.
How many nonzero entries does $A$ have, what are their locations, and what are the entries of $A^{-1}$ in terms of the nonnegative entries of $A$?
My attempt:
Diagonal matrices $D$ with all positive diagonal elements are examples of matrices so characterized. Such matrices have $n^2-n$ zero entries and $n$ non-zero entries and the non-zero elements of $D^{-1}$ are diagonal entries, each the reciprocal of the corresponding location on the diagonal of $D$.
For $A$ not diagonal, I tried equating cofactors of $A$ with cofactors of $A^{-1}$ and seeing which entries are equal to their negative. I couldn't get any obvious way to equate them, though.
Edit: I'd prefer a proof that doesn't rely on graph theory, a subject in which I'm unfamiliar. Presumably this is solvable just with matrix algebra or determinants.