Answer
$v_∞=[\frac{1}{3}~~\frac{2}{3}]$
Work Step by Step
$v_∞=[x~~y]$
It must satisfy:
$v_∞P=v_∞$
$[x~~y]\begin{bmatrix}
.2 & .8 \\ .4 & .6 \\ \end{bmatrix}
=[x~~y]$
It gives us two equations:
$0.2x+0.4y=x$
$0.4y=0.8x$
$y=2x$
and
$0.8x+0.6y=y$
$0.8x=0.4y$
$y=2x~~$ (But, it is the same equation)
Also:
$x+y=1$
$x+2x=1$
$3x=1$
$x=\frac{1}{3}$
$y=2x$
$y=2\times\frac{1}{3}=\frac{2}{3}$
Finally:
$v_∞=[\frac{1}{3}~~\frac{2}{3}]$
