International Mathematics Competition
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2020

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IMC2020: Day 1, Problem 2

Problem 2. Let \(\displaystyle A\) and \(\displaystyle B\) be \(\displaystyle n\times n\) real matrices such that

\(\displaystyle \textrm{rk}(AB-BA+I)=1 \)

where \(\displaystyle I\) is the \(\displaystyle n\times n\) identity matrix.

Prove that

\(\displaystyle \tr(ABAB)-\tr(A^2B^2)=\frac12 n(n-1). \)

(\(\displaystyle \textrm{rk}(M)\) denotes the rank of matrix \(\displaystyle M\), i.e., the maximum number of linearly independent columns in \(\displaystyle M\). \(\displaystyle \tr(M)\) denotes the trace of \(\displaystyle M\), that is the sum of diagonal elements in \(\displaystyle M\).)

Rustam Turdibaev, V. I. Romanovskiy Institute of Mathematics

Hint: Let \(\displaystyle X=AB-BA\). What is \(\displaystyle \tr(X^2)\)?

    

IMC
2020

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