I need help with a proof of all information is included and all part of the same question,

Why does Colleys Method always preserve the average rating of 1/2?

Colley's Method is

Where I is the identity matrix, and is a matrix added for each game k. Therefore if team i plays team j in the game, the matrix has the form , and , with all other enteries 0. For example if team 1 plays team 2 in the first game then

To show that the method always preserves the average rating of 1/2, we need to show where N is the number of teams. Given the Colley method Cr=b, consider

where is the element of the vector (Cr) and is the element of the vector b. Substituting Equation (1) in Equation (2), show that the method preserves the average rating of 1/2.

allgames C = 21 + E G* allgames C = 21 + E G*