Inequality for 0-1-matrices.

linear-algebra matrices inverse inequality

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

Related

Towards a Little proof of Fermat's last theorem
Proof of $\int_0^\infty\frac{\left (1- e^{\pi\sqrt3x}\cos(\pi x )\right )e^{-2\pi x/\sqrt3}}{x(1+x^3)(1+x^3/2^3)(1+x^3/3^3)\dots}~dx=0.$
How many different figures can be formed with a regular polygon of $n$ vertices and a number $d$ of diagonals of this polygon?
Pouring water from bottles
Detailed analysis of the secretary problem
If an infinite set $S$ of positive integers is equidistributed, is $S+S$ also equidistributed?
References for learning real analysis background for understanding the Atiyah--Singer index theorem
Existence of the limit $\lim_{h\to0} \frac{b^h-1}h$ without knowing $b^x$ is differentiable
Prove that $(-1)^n \text{Laguerre}_n(2) \leq 1$.
"Numerical results" from category theory
Elevator stops in building
proving that $\alpha_1+\alpha_2<-3$ which are the negative roots of $f(x)$