As anon points you out, try to use $\overline{z\cdot w} = \overline{z}\cdot \overline{w}$ and simplify notation.
For instance, you could just write vectors in $\mathbb{C}^2$ like this: $(z_1, z_2)$, with $z_1, z_2 \in \mathbb{C}$.
Second, I would try to convince myself that that product/conjugacy rule works too for products of matrices and vectors. That is:
$$
\overline{A\cdot v} = \overline{A}\cdot \overline{v} \ ,
$$
for, say, $A$ a $2\times 2$ complex matrix and $v\in \mathbb{C}^2$. What would happen if $A$ had real coefficitients?
Finally, I would write the equality I already know, namely
$$
A \cdot Y_0 = \lambda Y_0 \ .
$$
Staring at it should force inspiration to come. :-)
The Perron-Frobenius theorem is used in Google's Pagerank. It's one of the things that make sure that the algorithm works. Here it's explained why the the theorem is useful, you have a lot of information with easy explanations on Google.
Best Answer
$A$ has real entries so $\bar{A}=A$. Now just conjugate $Av=\lambda v$ you get that $\bar{\lambda}$ is an eigenvalue corresponding to $\bar{v}$. (Remember this is only true because $A$ has real entries).
So $-1-8i$ is an eigenvalue with eigenvector $(1+2i,-1,-8i)^T$