[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]

Define [math]\mat{P}[/math] and [math]\mat{y}[/math] by

[[math]] \mat {P} = \pmatrix{ .5 & .5 \cr.25 & .75 }, \qquad \mat {y} = \pmatrix{ 1 \cr 0 }\ . [[/math]]

Compute [math]\mat {P}\mat{y}[/math], [math]\mat {P}^2 \mat {y}[/math], and [math]\mat {P}^4 \mat {y}[/math] and show that the results are approaching a constant vector. What is this vector?