RO-NMO 2016 Problem 111

Let be a $2\times 2$ real matrix $A$ that has the property that $\left| A^d-I_2 \right| =\left| A^d+I_2 \right| ,$ for all $d\in\{ 2014,2016 \} .$ Prove that $\left| A^n-I_2 \right| =\left| A^n+I_2 \right| ,$ for any natural number $n.$