[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]

Show that if [math]\mat P[/math] is the transition matrix of a regular Markov chain, and [math]\mat W[/math] is the matrix each of whose rows is the fixed probability vector corresponding to [math]\mat {P}[/math], then [math]\mat {P}\mat {W} = \mat {W}[/math], and [math]\mat{W}^k = \mat {W}[/math] for all positive integers [math]k[/math].