Page 921 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

In this problem we develop a formula for whose -th entry equals the number of k-step connections from to .
(a) Use a computer to compute the eight matrices for , 3 and for , 3, 4, 5.
(b) Use the results in part (a) and symmetry arguments to show that can be written as

(c) Using the fact that , show that

with

(d) Using part (c), show that

(e) Use the methods of Section 7.2 to compute

and thereby obtain expressions for and , and eventually show that

where is the matrix all of whose entries are ones and is the identity matrix.

(f) Show that for , all vertices for these directed graphs belong to cliques.

Consider a round-robin tournament among n players (labeled , , , …, ) where beats , beats , beats ,
T2. …, beats , and beats . Compute the “power” of each player, showing that they all have the same power; then

determine that common power.

Hint Use a computer to study the cases , 4, 5, 6; then make a conjecture and prove your conjecture to be true.