Given a $n \times n$ matrix $A$ with entries 0 or 1 and non-zero determinant.
Question 1: Is true that the sum of the entries of the inverse of $A$ is less than or equal to $n$?
Question 2: Is true that the sum of the entries of the inverse of $A$ is less than or equal to $n$ in case $A$ has determinant $\pm 1$ and only the entries 1 on the diagonal? Is in this case the value $n$ uniquely attained by the identity matrix as this sum?
Both questions have a positive answer for $n \leq 4$. It would also be nice when someone with a good program/computer could gheck it for n=5 or even n=6.
Best Answer
You don't work my friend. Consider a few thousand random $10\times 10$ matrices and you can get $\sum_{i,j}A^{-1}_{i,j}\geq 16$.
EDIT. If I understand correctly your second point, you consider $0-1$ matrices with diagonal $1,\cdots, 1$ and $\det(A)=\pm 1$.
With a random test, I find the following $10\times 10$ matrix $A$ with $\det(A)=1$ and $\sum_{i,j}A^{-1}_{i,j}=21$.