Question: Let P be an \(n\times n\) probability matrix and let \(X={{[1,1,....,1]}^{T}}\) be the nx1 matrix, all whose entries are 1. What is \({{P}^{T}}X\) ? (Try some examples if you are not sure.) How does it follow that \(\det \left( {{P}^{T}}-I \right)\) =0?

How does it follow that \(\det \left( P-I \right)=0\) ? How does this relate to proving theorem 3?

{Theorem 3; let P be a probability matrix with no entries equal to 0. Then there is a unique probability vector X such that PX=X. If D is any probability vector, then (lim->∞ P^nD=X)}