The inverse of a symmetric nonnegative matrix can be copositive

Let $A$ be an $n\times n$ symmetric matrix. We first show that if $A$ and its pseudoinverse are strictly copositive, then $A$ is positive semidefinite, which extends a similar result of Han and Mangasarian. Suppose $A$ is invertible, as well as being symmetric. We showed in an earlier paper that if $A^{-1}$ is nonnegative with $n$ zero diagonal entries, then $A$ can be copositive (for instance, this happens with the Horn matrix), and when $A$ is copositive, it cannot be of form $P+N$, where $P$ is positive semidefinite and $N$ is nonnegative and symmetric. Here, we show that if $A^{-1}$ is nonnegative with $n-1$ zero diagonal entries and one positive diagonal entry, then $A$ can be of the form $P+N$, and we show how to construct $A$. We also show that if $A^{-1}$ is nonnegative with one zero diagonal entry and $n-1$ positive diagonal entries, then $A$ cannot be copositive.