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Romania District Olympiad 2001 - Grade XI

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March 16, 2011
linear algebramatrixlinear algebra unsolved

Problem Statement

Let n∈N, n≥2n\in \mathbb{N},\ n\ge 2. For any matrix A∈Mn(C)A\in \mathcal{M}_n(\mathbb{C}), let m(A)m(A) be the number of non-zero minors of AA. Prove that:
a)m(In)=2n−1m(I_n)=2^n-1; b)If A∈Mn(C)A\in \mathcal{M}_n(\mathbb{C}) is non-singular, then m(A)≥2n−1m(A)\ge 2^n-1.
Marius Ghergu
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