Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 61 / 2 / a / Solution

From $A^2=I$, induction gives $A^{2n}=I$ and $A^{2n+1}=A$. The matrix exponential converges absolutely in finite dimension, so its even and odd terms may be separated:This involution exponential formula requires no Hermiticity assumption on $A$. Hermiticity is needed if the exponential is additionally asserted to be unitary.