Eigenvectors/Eigenvalues help!

Hi everyone

I have a piece of maths coursework of an engineering course which the final part of which is providing some difficulties, so was wondering if someone here would be kind enough to help please.

I am aware of what eigenvalues/vectors are, but i am not certain how to apply the things i know to this question - nor does anyone on my course or any other doctor i have asked!

Anyway, the question (or the very first part!).

Show that:

$\displaystyle

\left[ \begin{array}{cccc} cos(x) & -sin(x) & 0 \\ sin(x)& cos(x) & 0 \\ 0 & 0 & 1 \end{array} \right]

$

Has the an eigenvector

$\displaystyle

\left[ \begin{array}{cccc} 1 \\ i \\ 0 \end{array} \right]

$

Write the answer in exponential form.

Id appreciate any help on the matter

thanks in advance!