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|A²+A+I|=|A²-A+I|=3 implies A²(A²+I)=2I

Source: Romanian District Olympiad 2016, Grade XI, Problem 1

October 5, 2018
linear algebraalgebraMatrices

Problem Statement

Let A∈M2(C) A\in M_2\left( \mathbb{C}\right) such that det⁡(A2+A+I2)=det⁡(A2−A+I2)=3. \det \left( A^2+A+I_2\right) =\det \left( A^2-A+I_2\right) =3. Prove that A2(A2+I2)=2I2. A^2\left( A^2+I_2\right) =2I_2.
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