Answer
$v_∞=[\frac{2}{3}~~\frac{1}{3}]$
Work Step by Step
$v_∞=[x~~y]$
It must satisfy:
$v_∞P=v_∞$
$[x~~y]\begin{bmatrix}
\frac{1}{2} & \frac{1}{2} \\
1 & 0 \\ \end{bmatrix}
=[x~~y]$
It gives us two equations:
$\frac{1}{2}x+y=x$
$y=\frac{1}{2}x$
and
$\frac{1}{2}x+0y=y~~$ (But, it is the same equation)
Also:
$x+y=1$
$x+\frac{1}{2}x=1$
$\frac{3}{2}x=1$
$x=\frac{2}{3}$
$y=\frac{1}{2}x$
$y=\frac{1}{2}\times\frac{2}{3}=\frac{1}{3}$
Finally:
$v_∞=[\frac{2}{3}~~\frac{1}{3}]$
