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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.

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    $\begingroup$ "Both questions have a positive answer for n≤4" . $\endgroup$ Commented Sep 30, 2019 at 15:47
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    $\begingroup$ @user247327 what do you mean with that quote? $\endgroup$ Commented Sep 30, 2019 at 15:48
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    $\begingroup$ "Both questions have a positive answer for n≤4" That's clearly NOT true for the matrix $\begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}$ a 2 by 2 matrix of 1s and 0s with determinant 1 sum of entries 3> 2. Did you mean that the sum of entries is less than or equal to $n^2$? (Why can comments only be edited for 5 seconds?) $\endgroup$ Commented Sep 30, 2019 at 15:56
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    $\begingroup$ @user247327 sum of entries of the inverse of that matrix (not the matrix itself)! $\endgroup$ Commented Sep 30, 2019 at 15:57
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    $\begingroup$ It isn't easy to prove. $\endgroup$ Commented Oct 1, 2019 at 13:52

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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 $\sum_{i,j}A^{-1}_{i,j}=21$.

enter image description here

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    $\begingroup$ Well, I would think that too, but maybe the second question is positive. Or do you have a concrete counterexample? $\endgroup$ Commented Oct 1, 2019 at 15:34
  • $\begingroup$ Can you give a concrete example for that? $\endgroup$ Commented Oct 1, 2019 at 16:40
  • $\begingroup$ Thanks. Do you also have such an example where the sum of all entries of the inverse is larger than the sum of all entries of the original matrix? See mathoverflow.net/questions/342871/inequality-for-0-1-matrices . $\endgroup$ Commented Oct 1, 2019 at 16:55

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