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IMC2021: Day 1, Problem 1Problem 1. Let \(\displaystyle A\) be a real \(\displaystyle n\times n\) matrix such that \(\displaystyle A^3=0\). (a) Prove that there is a unique real \(\displaystyle n\times n\) matrix \(\displaystyle X\) that satisfies the equation \(\displaystyle X+AX+XA^2=A. \) (b) Express \(\displaystyle X\) in terms of \(\displaystyle A\). Bekhzod Kurbonboev, Institute of Mathematics, Tashkent | |||||||||||||
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