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linear algebra problem with equality of determinants

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

August 25, 2019

linear algebramatrix

Problem Statement

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.