Problem

Source: Romanian National Olympiad 2016, grade xi, p.1

Tags: linear algebra, matrix



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. $