We consider $R \in \mathrm{O}_{d}(\mathbb{R})$.
\begin{itemize}
\item[(a)] Show that if $\lambda$ is an eigenvalue of $R$ then $\lambda \in \{+1, -1\}$.
\item[(b)] Show that $\operatorname{det}(R + I) = \operatorname{det}(R) \operatorname{det}(I + R^{T})$.
\item[(c)] Deduce that if $\operatorname{det}(R) = -1$ then $\operatorname{det}(R + I) = 0$.
\end{itemize}