Find the state vector $\textbf{x}_3$ for the given stochastic matrix A and initial state vector $\textbf{x}_0$.
$A = \left[\begin{array}{cc} {\frac{8}{11}} &{\frac{21}{23}}\cr {\frac{3}{11}} &{\frac{2}{23}} \end{array}\right]$,   $\textbf{x}_0= \left[\begin{array}{c} {\frac{4}{7}}\cr {\frac{3}{7}} \end{array}\right]$

$\textbf{x}_3 =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$