Suppose there are four teams in a curling league. At the end of the season, the results are as follows:
Team 1 beat teams 2 and 3, but lost to team 4.
Team 2 beat team 3, but lost to teams 1 and 4.
Team 3 beat team 4, but lost to teams 1 and 2.
Team 4 beat teams 1 and 2, but lost to team 3.
(a) Form the corresponding matrix A that reflects these results, where
aij = (1 if team i beat team j
0 otherwise
(b) How small can the dominant eigenvalue for A be? How large? Explain.
(c) It turns out that the dominant eigenvalue is approximately 1.395, and the corresponding eigenvector is
v =
0.552
0.321
0.448
0.626
How should the teams be ranked?