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 ?

The following characterization of rotational matrices can be helpful, especially for matrix size $n > 2$.
$M$ is a rotational matrix if and only if $M$ is orthogonal, i.e. $MM^T = M^TM = I$, and $\det(M) = 1$.
