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Romanian National Olympiad 2018 - Grade 11 - problem 4

Source: Romania NMO - 2018

April 7, 2018
linear algebraMatricesrankmatrix

Problem Statement

Let nn be an integer with n≥2n \geq 2 and let A∈Mn(C)A \in \mathcal{M}_n(\mathbb{C}) such that rank⁡A≠rank⁡A2.\operatorname{rank} A \neq \operatorname{rank} A^2. Prove that there exists a nonzero matrix B∈Mn(C)B \in \mathcal{M}_n(\mathbb{C}) such that AB=BA=B2=0AB=BA=B^2=0
Cornel Delasava
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