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IMC 2021, first day, problem 1

Source: IMC day1 , p1

August 4, 2021
linear algebraIMC 2021

Problem Statement

Let AA be a real n×nn\times n matrix such that A3=0A^3=0 a) prove that there is unique real n×nn\times n matrix XX that satisfied the equation X+AX+XA2=AX+AX+XA^2=A b) Express XX in terms of AA
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