State the conditions on a matrix $A$ that ensure it represents a rotation of $\mathbb{R}^{3}$ with respect to the standard basis.

Check that the matrix

$$A=\frac{1}{3}\left(\begin{array}{ccc} -1 & 2 & -2 \\ 2 & 2 & 1 \\ 2 & -1 & -2 \end{array}\right)$$

represents a rotation. Find its axis of rotation $\mathbf{n}$.

Let $\Pi$ be the plane perpendicular to the axis $\mathbf{n}$. The matrix $A$ induces a rotation of $\Pi$ by an angle $\theta$. Find $\cos \theta$.