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Antisymetric matrix

Source: RMO 2008, 11th Grade, Problem 4

April 30, 2008
linear algebramatrixalgebrapolynomialinequalitieslinear algebra unsolved

Problem Statement

Let A\equal{}(a_{ij})_{1\leq i,j\leq n} be a real n×n n\times n matrix, such that a_{ij} \plus{} a_{ji} \equal{} 0, for all i,j i,j. Prove that for all non-negative real numbers x,y x,y we have \det(A\plus{}xI_n)\cdot \det(A\plus{}yI_n) \geq \det (A\plus{}\sqrt{xy}I_n)^2.
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