Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 10 / a / Solution

The upper-left block of $U$ is the planar rotation matrix $R(\pi /2)$, and $U$ fixes the third coordinate. ThereforeThe diagonal entries of $V$ square to one, so $V^2=I$. Direct conjugation gives $VUV=U^{-1}$, hence $V{U}^{p}V=U^{-p}$ for every integer $p$. ConsequentlyWhen $p\mathrm{mod}4=0$, the last exponent is four, which is equivalent to zero because $U^4=I$.