Find all possible numbers that of positive elements in the inverse of an $n\times n$ positive matrix.
For $n=2$, that is only $2$. This is because the inverse of a $2\times2$ positive matrix is of the form of $\begin{pmatrix}+&-\\\ -&+\end{pmatrix}$ or $\begin{pmatrix}-&+\\\ +&-\end{pmatrix}$.
An upper limit is $n^2-n$. Write $B=A^{-1}$, then $AB$ is a unit matrix, and the inner product of each horizontal vector $a_i$ of $A$ and each vertical vector $b_j$ of $B$ is zero for $i\neq j$, but since all the elements of $A$ are positive, at least one of the elements of each vertical vector of $B$ must be negative. In the same way, we have a lower limit $n$. Also, I did some numerical computation for $n=3,4,5$ and it appears that all of them from $n$ to $n^2-n$ exist.
This is the result I got:
$n=3$ $$\begin{pmatrix}1 & 2 & 1\\2 & 1 & 1\\1 & 1 & 1\end{pmatrix}^{-1}=\begin{pmatrix}0 & \color{red}{1} & -1\\ \color{red}{1} & 0 & -1\\-1 & -1 & \color{red}{3}\end{pmatrix}$$ $$\begin{pmatrix}2 & 1 & 2\\2 & 2 & 1\\1 & 1 & 1\end{pmatrix}^{-1}=\begin{pmatrix} \color{red}{1} & \color{red}{1} & -3\\-1 & 0 & \color{red}{2}\\0 & -1 & \color{red}{2}\end{pmatrix}$$ $$\begin{pmatrix}3 & 2 & 1\\2 & 1 & 1\\1 & 1 & 1\end{pmatrix}^{-1}=\begin{pmatrix}0 & \color{red}{1} & -1\\ \color{red}{1} & -2 & \color{red}{1}\\-1 & \color{red}{1} & \color{red}{1}\end{pmatrix}$$ $$\begin{pmatrix}1 & 2 & 2\\2 & 1 & 2\\2 & 2 & 1\end{pmatrix}^{-1}=\frac{1}{5}\begin{pmatrix}-3 & \color{red}{2} & \color{red}{2}\\ \color{red}{2} & -3 & \color{red}{2}\\ \color{red}{2} & \color{red}{2} & -3\end{pmatrix}$$
$n=4$ $$\begin{pmatrix}1 & 1 & 2 & 1\\1 & 2 & 1 & 1\\2 & 1 & 1 & 1\\1 & 1 & 1 & 1\end{pmatrix}^{-1}=\begin{pmatrix}0 & 0 & \color{red}{1} & -1\\0 & \color{red}{1} & 0 & -1\\ \color{red}{1} & 0 & 0 & -1\\-1 & -1 & -1 & \color{red}{4}\end{pmatrix}$$ $$\begin{pmatrix}2 & 2 & 2 & 1\\1 & 2 & 1 & 1\\2 & 1 & 1 & 1\\1 & 1 & 1 & 1\end{pmatrix}^{-1}=\begin{pmatrix}0 & 0 & \color{red}{1} & -1\\0 & \color{red}{1} & 0 & -1\\ \color{red}{1} & -1 & -1 & \color{red}{1}\\-1 & 0 & 0 & \color{red}{2}\end{pmatrix}$$ $$\begin{pmatrix}1 & 2 & 1 & 2\\1 & 2 & 2 & 1\\2 & 1 & 1 & 1\\1 & 1 & 1 & 1\end{pmatrix}^{-1}=\begin{pmatrix}0 & 0 & \color{red}{1} & -1\\ \color{red}{1} & \color{red}{1} & \color{red}{1} & -4\\-1 & 0 & -1 & \color{red}{3}\\0 & -1 & -1 & \color{red}{3}\end{pmatrix}$$ $$\begin{pmatrix}1 & 2 & 2 & 1\\2 & 1 & 2 & 1\\2 & 2 & 1 & 1\\1 & 1 & 1 & 1\end{pmatrix}^{-1}=\frac{1}{2}\begin{pmatrix}-1 & \color{red}{1} & \color{red}{1} & -1\\ \color{red}{1} & -1 & \color{red}{1} & -1\\ \color{red}{1} & \color{red}{1} & -1 & -1\\-1 & -1 & -1 & \color{red}{5}\end{pmatrix}$$ $$\begin{pmatrix}2 & 2 & 1 & 2\\2 & 2 & 2 & 1\\1 & 2 & 1 & 1\\2 & 1 & 1 & 1\end{pmatrix}^{-1}=\begin{pmatrix}-1 & -1 & \color{red}{1} & \color{red}{2}\\-1 & -1 & \color{red}{2} & \color{red}{1}\\ \color{red}{1} & \color{red}{2} & -2 & -2\\ \color{red}{2} & \color{red}{1} & -2 & -2\end{pmatrix}$$ $$\begin{pmatrix}2 & 1 & 1 & 2\\2 & 1 & 2 & 1\\2 & 2 & 1 & 1\\1 & 1 & 1 & 1\end{pmatrix}^{-1}=\frac{1}{2}\begin{pmatrix} \color{red}{1} & \color{red}{1} & \color{red}{1} & -4\\-1 & -1 & \color{red}{1} & \color{red}{2}\\-1 & \color{red}{1} & -1 & \color{red}{2}\\ \color{red}{1} & -1 & -1 & \color{red}{2}\end{pmatrix}$$ $$\begin{pmatrix}3 & 1 & 2 & 3\\3 & 2 & 2 & 1\\2 & 2 & 1 & 1\\1 & 1 & 1 & 1\end{pmatrix}^{-1}=\frac{1}{3}\begin{pmatrix} \color{red}{1} & \color{red}{1} & \color{red}{1} & -5\\-1 & -1 & \color{red}{2} & \color{red}{2}\\-1 & \color{red}{2} & -4 & \color{red}{5}\\ \color{red}{1} & -2 & \color{red}{1} & \color{red}{1}\end{pmatrix}$$ $$\begin{pmatrix}2 & 1 & 3 & 2\\3 & 2 & 1 & 2\\3 & 2 & 2 & 1\\1 & 1 & 1 & 1\end{pmatrix}^{-1}=\frac{1}{4}\begin{pmatrix} \color{red}{1} & \color{red}{2} & \color{red}{1} & -7\\-3 & -2 & \color{red}{1} & \color{red}{9}\\ \color{red}{1} & -2 & \color{red}{1} & \color{red}{1}\\ \color{red}{1} & \color{red}{2} & -3 & \color{red}{1}\end{pmatrix}$$ $$\begin{pmatrix}1 & 2 & 2 & 2\\2 & 1 & 2 & 2\\2 & 2 & 1 & 2\\2 & 2 & 2 & 1\end{pmatrix}^{-1}=\frac{1}{7}\begin{pmatrix}-5 & \color{red}{2} & \color{red}{2} & \color{red}{2}\\ \color{red}{2} & -5 & \color{red}{2} & \color{red}{2}\\ \color{red}{2} & \color{red}{2} & -5 & \color{red}{2}\\ \color{red}{2} & \color{red}{2} & \color{red}{2} & -5\end{pmatrix}$$
$n=5$ $$\begin{pmatrix}1 & 1 & 1 & 2 & 1\\1 & 1 & 2 & 1 & 1\\1 & 2 & 1 & 1 & 1\\2 & 1 & 1 & 1 & 1\\1 & 1 & 1 & 1 & 1\end{pmatrix}^{-1}=\begin{pmatrix}0 & 0 & 0 & \color{red}{1} & -1\\0 & 0 & \color{red}{1} & 0 & -1\\0 & \color{red}{1} & 0 & 0 & -1\\ \color{red}{1} & 0 & 0 & 0 & -1\\-1 & -1 & -1 & -1 & \color{red}{5}\end{pmatrix}$$ $$\begin{pmatrix}2 & 2 & 1 & 2 & 1\\1 & 1 & 2 & 1 & 1\\1 & 2 & 1 & 1 & 1\\2 & 1 & 1 & 1 & 1\\1 & 1 & 1 & 1 & 1\end{pmatrix}^{-1}=\begin{pmatrix}0 & 0 & 0 & \color{red}{1} & -1\\0 & 0 & \color{red}{1} & 0 & -1\\0 & \color{red}{1} & 0 & 0 & -1\\ \color{red}{1} & 0 & -1 & -1 & \color{red}{1}\\-1 & -1 & 0 & 0 & \color{red}{3}\end{pmatrix}$$ $$\begin{pmatrix}2 & 2 & 1 & 2 & 1\\2 & 1 & 2 & 1 & 1\\1 & 2 & 1 & 1 & 1\\2 & 1 & 1 & 1 & 1\\1 & 1 & 1 & 1 & 1\end{pmatrix}^{-1}=\begin{pmatrix}0 & 0 & 0 & \color{red}{1} & -1\\0 & 0 & \color{red}{1} & 0 & -1\\0 & \color{red}{1} & 0 & -1 & 0\\ \color{red}{1} & 0 & -1 & -1 & \color{red}{1}\\-1 & -1 & 0 & \color{red}{1} & \color{red}{2}\end{pmatrix}$$ $$\begin{pmatrix}1 & 1 & 2 & 2 & 1\\2 & 2 & 2 & 1 & 1\\1 & 2 & 1 & 1 & 1\\2 & 1 & 1 & 1 & 1\\1 & 1 & 1 & 1 & 1\end{pmatrix}^{-1}=\begin{pmatrix}0 & 0 & 0 & \color{red}{1} & -1\\0 & 0 & \color{red}{1} & 0 & -1\\0 & \color{red}{1} & -1 & -1 & \color{red}{1}\\ \color{red}{1} & -1 & \color{red}{1} & \color{red}{1} & -2\\-1 & 0 & -1 & -1 & \color{red}{4}\end{pmatrix}$$ $$\begin{pmatrix}2 & 2 & 1 & 2 & 1\\2 & 2 & 2 & 1 & 1\\1 & 2 & 1 & 1 & 1\\2 & 1 & 1 & 1 & 1\\1 & 1 & 1 & 1 & 1\end{pmatrix}^{-1}=\begin{pmatrix}0 & 0 & 0 & \color{red}{1} & -1\\0 & 0 & \color{red}{1} & 0 & -1\\0 & \color{red}{1} & -1 & -1 & \color{red}{1}\\ \color{red}{1} & 0 & -1 & -1 & \color{red}{1}\\-1 & -1 & \color{red}{1} & \color{red}{1} & \color{red}{1}\end{pmatrix}$$ $$\begin{pmatrix}2 & 2 & 1 & 1 & 2\\1 & 2 & 1 & 2 & 1\\1 & 2 & 2 & 1 & 1\\2 & 1 & 1 & 1 & 1\\1 & 1 & 1 & 1 & 1\end{pmatrix}^{-1}=\frac{1}{2}\begin{pmatrix}0 & 0 & 0 & \color{red}{2} & -2\\ \color{red}{1} & \color{red}{1} & \color{red}{1} & 0 & -4\\-1 & -1 & \color{red}{1} & 0 & \color{red}{2}\\-1 & \color{red}{1} & -1 & 0 & \color{red}{2}\\ \color{red}{1} & -1 & -1 & -2 & \color{red}{4}\end{pmatrix}$$ $$\begin{pmatrix}1 & 2 & 1 & 1 & 2\\1 & 2 & 1 & 2 & 1\\1 & 2 & 2 & 1 & 1\\2 & 1 & 1 & 1 & 1\\1 & 1 & 1 & 1 & 1\end{pmatrix}^{-1}=\frac{1}{2}\begin{pmatrix}0 & 0 & 0 & \color{red}{2} & -2\\ \color{red}{1} & \color{red}{1} & \color{red}{1} & \color{red}{1} & -5\\-1 & -1 & \color{red}{1} & -1 & \color{red}{3}\\-1 & \color{red}{1} & -1 & -1 & \color{red}{3}\\ \color{red}{1} & -1 & -1 & -1 & \color{red}{3}\end{pmatrix}$$ $$\begin{pmatrix}2 & 1 & 1 & 1 & 2\\2 & 1 & 1 & 2 & 1\\2 & 1 & 2 & 1 & 1\\2 & 2 & 1 & 1 & 1\\1 & 1 & 1 & 1 & 1\end{pmatrix}^{-1}=\frac{1}{3}\begin{pmatrix} \color{red}{1} & \color{red}{1} & \color{red}{1} & \color{red}{1} & -5\\-1 & -1 & -1 & \color{red}{2} & \color{red}{2}\\-1 & -1 & \color{red}{2} & -1 & \color{red}{2}\\-1 & \color{red}{2} & -1 & -1 & \color{red}{2}\\ \color{red}{2} & -1 & -1 & -1 & \color{red}{2}\end{pmatrix}$$ $$\begin{pmatrix}1 & 2 & 2 & 1 & 2\\1 & 2 & 2 & 2 & 1\\2 & 1 & 2 & 1 & 1\\2 & 2 & 1 & 1 & 1\\1 & 1 & 1 & 1 & 1\end{pmatrix}^{-1}=\frac{1}{3}\begin{pmatrix}-1 & -1 & \color{red}{1} & \color{red}{1} & \color{red}{1}\\ \color{red}{1} & \color{red}{1} & -1 & \color{red}{2} & -4\\ \color{red}{1} & \color{red}{1} & \color{red}{2} & -1 & -4\\-2 & \color{red}{1} & -1 & -1 & \color{red}{5}\\ \color{red}{1} & -2 & -1 & -1 & \color{red}{5}\end{pmatrix}$$ $$\begin{pmatrix}2 & 2 & 2 & 1 & 2\\2 & 1 & 2 & 2 & 1\\2 & 2 & 1 & 2 & 1\\1 & 2 & 2 & 1 & 1\\2 & 1 & 1 & 1 & 1\end{pmatrix}^{-1}=\frac{1}{3}\begin{pmatrix}-2 & -1 & -1 & \color{red}{1} & \color{red}{5}\\-1 & -2 & \color{red}{1} & \color{red}{2} & \color{red}{1}\\-1 & \color{red}{1} & -2 & \color{red}{2} & \color{red}{1}\\ \color{red}{1} & \color{red}{2} & \color{red}{2} & -2 & -4\\ \color{red}{5} & \color{red}{1} & \color{red}{1} & -4 & -5\end{pmatrix}$$ $$\begin{pmatrix}2 & 1 & 1 & 2 & 2\\2 & 1 & 2 & 1 & 2\\2 & 1 & 2 & 2 & 1\\2 & 2 & 1 & 1 & 1\\1 & 1 & 1 & 1 & 1\end{pmatrix}^{-1}=\frac{1}{3}\begin{pmatrix} \color{red}{1} & \color{red}{1} & \color{red}{1} & \color{red}{2} & -7\\-1 & -1 & -1 & \color{red}{1} & \color{red}{4}\\-2 & \color{red}{1} & \color{red}{1} & -1 & \color{red}{2}\\ \color{red}{1} & -2 & \color{red}{1} & -1 & \color{red}{2}\\ \color{red}{1} & \color{red}{1} & -2 & -1 & \color{red}{2}\end{pmatrix}$$ $$\begin{pmatrix}3 & 1 & 1 & 3 & 2\\3 & 2 & 2 & 1 & 2\\2 & 1 & 2 & 2 & 1\\2 & 2 & 1 & 1 & 1\\1 & 1 & 1 & 1 & 1\end{pmatrix}^{-1}=\frac{1}{5}\begin{pmatrix} \color{red}{1} & \color{red}{2} & \color{red}{1} & \color{red}{1} & -8\\-1 & -2 & -1 & \color{red}{4} & \color{red}{3}\\-2 & \color{red}{1} & \color{red}{3} & -2 & \color{red}{1}\\ \color{red}{1} & -3 & \color{red}{1} & \color{red}{1} & \color{red}{2}\\ \color{red}{1} & \color{red}{2} & -4 & -4 & \color{red}{7}\end{pmatrix}$$ $$\begin{pmatrix}1 & 2 & 2 & 2 & 2\\2 & 1 & 2 & 2 & 2\\2 & 2 & 1 & 1 & 2\\2 & 2 & 1 & 2 & 1\\2 & 2 & 2 & 1 & 1\end{pmatrix}^{-1}=\frac{1}{12}\begin{pmatrix}-8 & \color{red}{4} & \color{red}{2} & \color{red}{2} & \color{red}{2}\\ \color{red}{4} & -8 & \color{red}{2} & \color{red}{2} & \color{red}{2}\\ \color{red}{2} & \color{red}{2} & -5 & -5 & \color{red}{7}\\ \color{red}{2} & \color{red}{2} & -5 & \color{red}{7} & -5\\ \color{red}{2} & \color{red}{2} & \color{red}{7} & -5 & -5\end{pmatrix}$$ $$\begin{pmatrix}1 & 3 & 2 & 2 & 3\\2 & 3 & 1 & 2 & 2\\2 & 3 & 2 & 2 & 1\\3 & 2 & 2 & 1 & 1\\1 & 1 & 1 & 1 & 1\end{pmatrix}^{-1}=\frac{1}{6}\begin{pmatrix}-2 & \color{red}{2} & -2 & \color{red}{2} & \color{red}{2}\\ \color{red}{2} & \color{red}{1} & \color{red}{2} & \color{red}{1} & -11\\ \color{red}{2} & -5 & \color{red}{2} & \color{red}{1} & \color{red}{1}\\-4 & \color{red}{1} & \color{red}{2} & -5 & \color{red}{13}\\ \color{red}{2} & \color{red}{1} & -4 & \color{red}{1} & \color{red}{1}\end{pmatrix}$$ $$\begin{pmatrix}2 & 1 & 3 & 2 & 2\\3 & 2 & 1 & 2 & 2\\3 & 2 & 2 & 1 & 2\\3 & 2 & 2 & 2 & 1\\1 & 1 & 1 & 1 & 1\end{pmatrix}^{-1}=\frac{1}{5}\begin{pmatrix} \color{red}{1} & \color{red}{2} & \color{red}{1} & \color{red}{1} & -9\\-4 & -3 & \color{red}{1} & \color{red}{1} & \color{red}{11}\\ \color{red}{1} & -3 & \color{red}{1} & \color{red}{1} & \color{red}{1}\\ \color{red}{1} & \color{red}{2} & -4 & \color{red}{1} & \color{red}{1}\\ \color{red}{1} & \color{red}{2} & \color{red}{1} & -4 & \color{red}{1}\end{pmatrix}$$ $$\begin{pmatrix}1 & 2 & 2 & 2 & 2\\2 & 1 & 2 & 2 & 2\\2 & 2 & 1 & 2 & 2\\2 & 2 & 2 & 1 & 2\\2 & 2 & 2 & 2 & 1\end{pmatrix}^{-1}=\frac{1}{9}\begin{pmatrix}-7 & \color{red}{2} & \color{red}{2} & \color{red}{2} & \color{red}{2}\\ \color{red}{2} & -7 & \color{red}{2} & \color{red}{2} & \color{red}{2}\\ \color{red}{2} & \color{red}{2} & -7 & \color{red}{2} & \color{red}{2}\\ \color{red}{2} & \color{red}{2} & \color{red}{2} & -7 & \color{red}{2}\\ \color{red}{2} & \color{red}{2} & \color{red}{2} & \color{red}{2} & -7\end{pmatrix}$$
