International Mathematics Competition
for University Students
2021

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IMC 2021
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IMC2021: Day 1, Problem 1

Problem 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

        

IMC
2021

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