Past exam of the mathematical methods I course of the University of Oxford / 2014 / 8

Here $M^2=-I$, so its even powers alternate between $\pm I$ and its odd powers between $\pm M$. Collecting even and odd terms of the matrix exponential yieldsThus $R^{T}R=I$, $\det R=1$, and $R(\varphi )(1,0{)}^T=(\cos \varphi ,\sin \varphi {)}^T$. It is the rotation matrix for a counterclockwise rotation through angle $\varphi$, preserving lengths and orientation.