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3x3 adjugate matrix

Source: RomNO 2003, grade xi, p.4

August 27, 2019
linear algebramatrixalgebrapolynomialadjugate

Problem Statement

Let be a 3×3 3\times 3 real matrix A. A. Prove the following statements. a) f(A)≠O3, f(A)\neq O_3, for any polynomials f∈R[X] f\in\mathbb{R} [X] whose roots are not real. b) \exists n\in\mathbb{N}  \left( A+\text{adj} (A) \right)^{2n} =\left( A \right)^{2n} +\left( \text{adj} (A) \right)^{2n}\iff \text{det} (A)=0
Laurențiu Panaitopol
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