Answer
$v_∞=[\frac{3}{7}~~\frac{4}{7}]$
Work Step by Step
$v_∞=[x~~y]$
It must satisfy:
$v_∞P=v_∞$
$[x~~y]\begin{bmatrix}
\frac{1}{3} & \frac{2}{3} \\ \frac{1}{2} & \frac{1}{2} \\ \end{bmatrix}
=[x~~y]$
It gives us two equations:
$\frac{1}{3}x+\frac{1}{2}y=x$
$\frac{1}{2}y=\frac{2}{3}x$
$y=\frac{4}{3}x$
and
$\frac{2}{3}x+\frac{1}{2}y=y$
$\frac{2}{3}x=\frac{1}{2}y$
$y=\frac{4}{3}x~~$ (But, it is the same equation)
Also:
$x+y=1$
$x+\frac{4}{3}x=1$
$\frac{7}{3}x=1$
$x=\frac{3}{7}$
$y=\frac{4}{3}x$
$y=\frac{4}{3}\times\frac{3}{7}=\frac{4}{7}$
Finally:
$v_∞=[\frac{3}{7}~~\frac{4}{7}]$
