A matrix for a round-robin tournament involving players competing in matches (no ties allowed) having entries

(1)

This scoring system differs from that used to compute a score sequence of a tournament, in which a win gives one point and a loss zero points.
The matrix satisfies

(2)

where
is the transpose of (McCarthy and Benjamin 1996).

The tournament matrix for players has zero determinantiff is odd (McCarthy and Benjamin
1996). Furthermore, the dimension of the null
space of an -player
tournament matrix is

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. Monthly102,
637-639, 1995.