[Math] Invertibility of a square matrix with zero diagonal elements and positive non-diagonal elements

linear algebramatricessingularity-theory

$M$ is square and
$$M(i,j)=0, i=j$$
$$M(i,j)>0, i\ne j$$
Is $M$ full-rank or invertible?

Actually the $M$ I am studying has much stronger properties but I guess the simple conditions above might be enough to make $M$ non-singular. The stronger properties of $M$ are:

  1. All elements in $M$ are non-negative integers between $0$ and $N$;

  2. The sum of each row is equal to $N$.

Edit

$N$ is not the dimension of $M$. It's just a constant positive integer.

I did search for this problem, but it seemed there was no much work on such matrices. It looks easy, but I don't know how to prove it and I couldn't find a counter-example either.

Best Answer

The matrix $M$ doesn't have to invertible. I give a simple counterexample here. $$M=\begin{pmatrix} 0&2&1&1\\ 2&0&1&1\\ 1&1&0&2\\ 1&1&2&0 \end{pmatrix}$$ You will see $r_1+r_2-r_3-r_4=0$ which indicates they are linear dependent.

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