[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 the matrix

[[math]] \mat {P} = \pmatrix{ 1 & 0 & 0 \cr 1/4 & 1/2 & 1/4 \cr 0 & 0 & 1} [[/math]]

has more than one fixed probability vector. Find the matrix that [math]\mat {P}^n[/math] approaches as [math]n \to \infty[/math], and verify that it is not a matrix all of whose rows are the same.