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Calculate PageRank for: A links to B, B links to C and C links to B and C where the damping factor $\beta=0.8$

I have:

$M=\begin{bmatrix} 0&0&\frac{1}{2} \\ 1&0&\frac{1}{2} \\ 0&1&0 \end{bmatrix}$

and

$v=\begin{bmatrix} \frac{1}{3} \\ \frac{1}{3} \\ \frac{1}{3} \end{bmatrix}$

So we get $(1-\beta)\frac{e}{n}=\begin{bmatrix} \frac{1}{15} \\ \frac{1}{15} \\ \frac{1}{15} \end{bmatrix}$

and $\beta M = \begin{bmatrix} 0&0&\frac{1}{30} \\ \frac{1}{15}&0&\frac{1}{30} \\ 0&\frac{1}{15}&0 \end{bmatrix}$

And therefore we get $$ v' =\begin{bmatrix} \frac{1}{15} \\ \frac{1}{15} \\ \frac{1}{15} \end{bmatrix}+\begin{bmatrix} 0&0&\frac{1}{30} \\ \frac{1}{15}&0&\frac{1}{30} \\ 0&\frac{1}{15}&0 \end{bmatrix} v $$

Is this correct?

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