[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Prove that, if a 3-by-3 transition matrix has the property that its column sums are 1, then [math](1/3, 1/3, 1/3)[/math] is a fixed probability vector. State a similar result for [math]n[/math]-by-[math]n[/math] transition matrices. Interpret these results for ergodic chains.