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Romanian District Olympiad 2019 - Grade 11 - Problem 2

Source: Romanian District Olympiad 2019 - Grade 11 - Problem 2

March 16, 2019
characteristic polynomialInequalityDeterminantslinear algebra

Problem Statement

Let n∈N,n≥2,n \in \mathbb{N},n \ge 2, and A,B∈Mn(R).A,B \in \mathcal{M}_n(\mathbb{R}). Prove that there exists a complex number z,z, such that ∣z∣=1|z|=1 and ℜ(det⁡(A+zB))≥det⁡(A)+det⁡(B),\Re \left( {\det(A+zB)} \right) \ge \det(A)+\det(B), where ℜ(w)\Re(w) is the real part of the complex number w.w.
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