  Let $G$ be a bipartite graph and its adjacency matrix $\mathbb A$. If $G$ has
a unique perfect matching, then $\mathbb A$ has an inverse $\mathbb A^{-1}$
which is a symmetric integral matrix, and hence the adjacency matrix of a
multigraph. The inverses of bipartite graphs with unique perfect matchings have
a strong connection to M\"obius functions of posets. In this note, we
characterize all bipartite graphs with a unique perfect matching whose
adjacency matrices have inverses diagonally similar to non-negative matrices,
which settles an open problem of Godsil on inverses of bipartite graphs in
[Godsil, Inverses of Trees, Combinatorica 5 (1985) 33-39].
