Past exam of the mathematics course of the University of Cambridge / 2004 / ia / Paper 3 / 6D (Algebra and Geometry)

A real three-by-three matrix represents a proper rotation in the standard orthonormal basis precisely when $A^{T}A=I$ and $\det A=1$, that is, when $A\in \mathrm{SO}(3)$. For the supplied integer numerator $B=3A$, direct multiplication gives $B^{T}B=9I$ and $\det B=27$, verifying both conditions.

The rotation axis is the eigenspace for eigenvalue one. Solving $(A-I)n=0$ gives $n_3=0$ and $n_2=2n_1$, sois a unit axis vector; its negative describes the same axis. On its perpendicular plane the rotation matrix has eigenvalues $e^{i\theta }$ and $e^{-i\theta }$. The rotation trace formula therefore gives $\mathrm{tr}A=1+2\cos \theta$. Since the trace is $-1/3$,