Let $n$ be a fixed positive integer. I would like to know for what values of $k$ there exists an $n$ by $n$ $0/1$ matrix that is non-singular with exactly $k$ ones per row.
Clearly if $k=1$ then the identity matrix is non-singular. Also if $k=n$ there are no non-singular matrices.
What can one say about $1 < k < n$?
If $n$ is large, is it true for almost all $k$ in the range?