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Antisymmetric matrix problem

Source: Romanian District Olympiad 2024 11.3

March 10, 2024
linear algebramatrix

Problem Statement

Let A∈Mn(C)A\in\mathcal{M}_n(\mathbb{C}) be an antisymmetric matrix, i.e. A=−At.A=-A^t. [*]Prove that if A∈Mn(R)A\in\mathcal{M}_n(\mathbb{R}) and A2=OnA^2=O_n then A=On.A=O_n. [*]Assume that nn{} is odd. Prove that if AA{} is the adjoint of another matrix B∈Mn(C)B\in\mathcal{M}_n(\mathbb{C}) then A2=On.A^2=O_n.
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