Theorem 1   The characteristic polynomial of a orthogonal matrix is a reciprocal polynomial

Proof. Let $A$ be the orthogonal matrix, and let $p(\lambda) = \det(A-\lambda I)$ be its characteristic polynomial. We wish to prove that $$ p(\lambda) = \pm \lambda^n p(1/\lambda). $$ Since $A^{-1}=A^T$ , we have $A-\lambda I=-\lambda A (A^T-I/\lambda ).$ Taking the determinant of both sides, and using $\det A = \det A^T$ and $\det c A = c^n \det A$ ($c\in \mathbb{C}$ ), yields $$ \det (A-\lambda I) = \pm \lambda^n \det(A-\frac{1}{\lambda} I).$$