How do I prove that a matrix is a rotation-matrix? I have to prove that this matrix is a rotation-matrix
$$\begin{pmatrix} \frac12 & 0 & \frac{\sqrt{3}}{2} \\
0 & 1 & 0 \\
\frac{\sqrt{3}}{2} & 0 & \frac12
\end{pmatrix}$$
How do I do this? 
My idea is to multiplicate it with $\begin{pmatrix} x \\ y \\ z\end{pmatrix}$ and show that one component will remain unchanged . Is this enough? Do non-rotational transformations exist, which leave one component unchanged ?

I think there is a minus sign missing. As it is, the determinant is not $1$. After fixing, this specific case is easy.
$$\begin{pmatrix} \frac12 & 0 & -\frac{\sqrt{3}}{2} \\
0 & 1 & 0 \\
\frac{\sqrt{3}}{2} & 0 & \frac12
\end{pmatrix} = \begin{pmatrix} \cos \frac{\pi}{3} & 0 & -\sin \frac{\pi}{3} \\ 0 & 1 & 0 \\ \sin \frac{\pi}{3} & 0 & \cos \frac{\pi}{3} \end{pmatrix}$$
It is a rotation of $\pi/3$ around the $y$-axis.
