Tournament Matrix
A matrix for a round-robin tournament involving 
players competing in 
matches (no ties allowed) having entries
The Matrix satisfies
where
is the Identity Matrix, 
is an 
Matrix of all 1s, and
is the Matrix Transpose of A. 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
(McCarthy 1996).
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.
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