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Romania NMO 2023 Grade 11 P3

Source: Romania National Olympiad 2023

April 14, 2023
linear algebraNilpotentrank

Problem Statement

Let nn be a natural number n≥2n \geq 2 and matrices A,B∈Mn(C),A,B \in M_{n}(\mathbb{C}), with property A2B=A.A^2 B = A.
a) Prove that (AB−BA)2=On.(AB - BA)^2 = O_{n}.
b) Show that for all natural number kk, k≤n2k \leq \frac{n}{2} there exist matrices A,B∈Mn(C)A,B \in M_{n}(\mathbb{C}) with property stated in the problem such that rank(AB−BA)=k.rank(AB - BA) = k.
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