Answer
$v_∞=[\frac{1}{5}~~\frac{4}{5}]$
Work Step by Step
$v_∞=[x~~y]$
It must satisfy:
$v_∞P=v_∞$
$[x~~y]\begin{bmatrix}
0 & 1 \\ \frac{1}{4} & \frac{3}{4} \\ \end{bmatrix}
=[x~~y]$
It gives us two equations:
$0x+\frac{1}{4}y=x$
$y=4x$
and
$x+\frac{3}{4}y=y~~$
$x=\frac{1}{4}y~~$ (But, it is the same equation)
Also:
$x+y=1$
$x+4x=1$
$5x=1$
$x=\frac{1}{5}$
$y=4x$
$y=4\times\frac{1}{5}=\frac{4}{5}$
Finally:
$v_∞=[\frac{1}{5}~~\frac{4}{5}]$
