[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]

Find the matrices [math]\mat{ P}^2,~\mat {P}^3,~\mat {P}^4,[/math]

and [math] \mat {P}^n[/math] for the Markov chain determined by the transition matrix [math] \mat {P} = \pmatrix{ 1 & 0 \cr 0 & 1 \cr}[/math]. Do the same for the transition matrix [math] \mat {P} = \pmatrix{ 0 & 1 \cr 1 & 0 \cr}[/math]. Interpret what happens in each of these processes.