A matrix for a round-robin tournament involving players competing in matches (no ties allowed)
having entries
The tournament matrix for players has zero Determinant Iff is Odd (McCarthy and Benjamin 1996).
The dimension of the Nullspace of an -player tournament matrix is
References
McCarthy, C. A. and Benjamin, A. T. ``Determinants of the Tournaments.'' Math. Mag. 69, 133-135, 1996.
Michael, T. S. ``The Ranks of Tournament Matrices.'' Amer. Math. Monthly 102, 637-639, 1995.