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Nice linear algebra

Source: Romanian District Olympiad 2023 11.4

March 11, 2023
linear algebramatrixrank

Problem Statement

Let AA{} and BB{} be 3×33\times 3{} matrices with complex entries, satisfying A2=B2=O3A^2=B^2=O_3. Prove that if AA{} and BB{} commute, then AB=O3AB=O_3. Is the converse true?
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