# International Mathematics Competition for University Students 2020

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IMC 2021

## 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 $\displaystyle \tr(X^2)$?

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