[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]

Let

[[math]] \mat {P} = \pmatrix{ 1 & 0 & 0 \cr .25 & 0 & .75 \cr 0 & 0 & 1 } [[/math]]

be a transition matrix of a Markov chain. Find two fixed vectors of [math]\mat {P}[/math] that are linearly independent. Does this show that the Markov chain is not regular?