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 determinant iff 
is odd (McCarthy and Benjamin
1996). Furthermore, the dimension of the null
space of an 
-player
tournament matrix is
![dim[nullspace]={0 for n even; 1 for n odd](/images/equations/TournamentMatrix/NumberedEquation3.svg) |
(3)
|
(McCarthy and Benjamin 1996).
