Answer
$v_∞=[\frac{7}{15}~~\frac{8}{15}]$
Work Step by Step
$v_∞=[x~~y]$
It must satisfy:
$v_∞P=v_∞$
$[x~~y]\begin{bmatrix}
.2 & .8 \\ .7 & .3 \\ \end{bmatrix}
=[x~~y]$
It gives us two equations:
$0.2x+0.7y=x$
$0.7y=0.8x$
$y=\frac{8}{7}x$
and
$0.8x+0.3y=y$
$0.8x=0.7y$
$y=\frac{8}{7}x~~$ (But, it is the same equation)
Also:
$x+y=1$
$x+\frac{8}{7}x=1$
$\frac{15}{7}x=1$
$x=\frac{7}{15}$
$y=\frac{3}{2}x$
$y=\frac{8}{7}\times\frac{7}{15}=\frac{8}{15}$
Finally:
$v_∞=[\frac{7}{15}~~\frac{8}{15}]$
