Question 61
Let $A \in \mathcal{M}_2(\mathbb{R})$ be a matrix satisfying the conditions:
$$
\det (A^{2014} - I_2) = \det (A^{2014} + I_2) \quad \text{and} \quad \det (A^{2016} - I_2) = \det (A^{2016} + I_2).
$$
Prove that $\det (A^n - I_2) = \det (A^n + I_2)$, for any positive integer $n$.
$$
\det (A^{2014} - I_2) = \det (A^{2014} + I_2) \quad \text{and} \quad \det (A^{2016} - I_2) = \det (A^{2016} + I_2).
$$
Prove that $\det (A^n - I_2) = \det (A^n + I_2)$, for any positive integer $n$.