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.

entriesof $A^{-1}$ or of theirabsolute values? The absolute values can grow exponentially; I gave an example at mathoverflow.net/questions/157472/… which can be taken to be upper triangular by permuting the rows or columns. $\endgroup$