International Mathematics Competition
for University Students
2021

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

Hint: (a) Multiply the equation by some power of \(\displaystyle A\) from left and another power of \(\displaystyle A\) from right.

(b) Substitute repeatedly \(\displaystyle X=A-AX-XA^2\).

  Solution