$M$ 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:
All elements in $M$ are non-negative integers between $0$ and $N$;
The sum of each row is equal to $N$.
$N$ 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.