[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]

Consider the Markov chain with transition matrix of Exercise, with [math]a = 0[/math] and [math]b = 1/2[/math]. Compute directly the unique fixed probability vector, and use your result to prove that the chain is not ergodic.