Given an $n \times n$-matrix $A$ with entries only $0$ or $1$ and determinant equal to $\pm 1$. Define the magnitude $M_A$ of $A$ as the sum of all entries of the inverse of $A$.
Question 0: Do we have that the sum of all entries of $A$ is greater than or equal to $M_A$ when $A$ additionally has all diagonal entries equal to one?
This is true for $n \leq 4$. One might also consider other bounds such as whether $M_A \leq n^2$.
Question 1: What is the maximal possible magnitude of such an $A$?
Question 2: What is the maximal value of the magnitude of $A$ in case it has additional all diagonal entries equal to $1$?
This is a small variant of a question asked on Math SE. The zeroth and second question is related to bounds for the magnitude of a Schurian algebra.