International Mathematics Competition
for University Students
2020

Select Year:


IMC 2023
Information
  Schedule
  Problems & Solutions
  Results
  Contact
 

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

© IMC