TITLE: Finding zero-one vectors in the row space of a matrix
QUESTION [5 upvotes]: Suppose that $M$ is a square matrix with all elements on its main diagonal equal to $1$, and every row containing exactly two off-diagonal elements equal to $-1$; all other elements are equal to $0$. The kernel of $M$ is nonzero and, indeed, contains a vector with all its coordinates nonzero. Does it follow that the row space of $M$ contains a (nonzero) zero-one vector?
In case it matters, the sum of all elements in every column of $M$ is nonpositive, and $M_{ij}M_{ji}=0$ whenever $i\ne j$.

REPLY [3 votes]: This is true:

For any square matrix $M$ with all elements on its main diagonal equal to 1, and every row containing exactly two off-diagonal elements equal to −1 (with all other elements are equal to 0), the row space of $M$ contains a nonzero zero-one vector. Moreover, there is a linear combination of the rows with the coefficients $0$ and $-1$ only which yields such a vector.

This is in fact the main lemma of this preprint; see, on the other hand, this MO problem for the explanations.