Answer
$v_∞=[\frac{2}{5}~~\frac{3}{5}]$
Work Step by Step
$v_∞=[x~~y]$
It must satisfy:
$v_∞P=v_∞$
$[x~~y]\begin{bmatrix}
.1 & .9 \\ .6 & .4 \\ \end{bmatrix}
=[x~~y]$
It gives us two equations:
$0.1x+0.6y=x$
$0.6y=0.9x$
$y=\frac{3}{2}x$
and
$0.9x+0.4y=y$
$0.9x=0.6y$
$y=\frac{3}{2}x~~$ (But, it is the same equation)
Also:
$x+y=1$
$x+\frac{3}{2}x=1$
$\frac{5}{2}x=1$
$x=\frac{2}{5}$
$y=\frac{3}{2}x$
$y=\frac{3}{2}\times\frac{2}{5}=\frac{3}{5}$
Finally:
$v_∞=[\frac{2}{5}~~\frac{3}{5}]$
