Answer
$v_\infty=\begin{bmatrix} \dfrac{1}{2}&\dfrac{1}{2}&0\end{bmatrix}$
Work Step by Step
The steady-state distribution vector $v_\infty$ can be written as: $v_\infty P=v_∞$
where, $v_\infty=[x~~y~~z]$
This gives: $[x~~y~~z] \begin{bmatrix}
1/2 & 1/2&0 \\ 1/2& 1/2&0\\1/2&0&1/2 \\ \end{bmatrix}
=[x~~y~~z]$
We can have the following equations:
$$(1/2)x+(1/2)y+(1/2)z=x\\ (1/2) x+(1/2) y=y\\\dfrac{1}{2} z=z$$
or, $$(-1/2)x+(1/2)y+(1/2)z=x\\ (1/2) x-(1/2) y=y\\ (-\dfrac{1}{2}) z=0$$
Also, we have: $x+y+z=1$
So, the new system of equations are:
$x+y+z=1\\ (1/2) x-(1/2) y=y\\ (-\dfrac{1}{2}) z=0 $
After solving the above equations, we get:
$x=1/2; y=1/2 ; z=0$
Thus, the required steady-state distribution vector $v_\infty$
$v_\infty=\begin{bmatrix} \dfrac{1}{2}&\dfrac{1}{2}&0\end{bmatrix}$
