Exercise 9.B.2

Proof. This basically the same as the previous exercise. Every operator on an odd-dimensional vector space has $n$-by-$n$ matrix for some odd integer $n$, hence one of the diagonal blocks is a $1$-by-$1$ matrix containing $1$ or $-1$. The basis vector that corresponds to column where this vector appears is an eigenvector corresponding to $1$ or $-1$. □