Let $A = \left[\begin{array}{ccc} 1 & 0 & 1/3 \\ 0 & 1 & 1/3 \\ 0 & 0 & 1/3 \end{array}\right]$
Numerically verify that each initial state vector $x_0$ has the given steady-state vector x.
$x_0 = \left[\begin{array}{c} 0.11\cr 0.29\cr 0.6 \end{array}\right]\Rightarrow \left[\begin{array}{c} 0.41\cr 0.59\cr 0 \end{array}\right]$
$x_1 = \left[\begin{array}{c} 0.1\cr 0.4\cr 0.5 \end{array}\right]\Rightarrow \left[\begin{array}{c} 0.35\cr 0.65\cr 0 \end{array}\right]$

$Ax_0 =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$ $Ax_1 =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$

$A^4x_0 =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$ $A^4x_1 =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$

$A^8x_0 =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$ $A^8x_1 =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$