Olympiad problems

2016 Korea USCM, 6

Tags: trace , commutator , common eigenvector , determinant , linear algebra , matrix

$A$ and $B$ are $2\times 2$ real valued matrices satisfying $$\det A = \det B = 1,\quad \text{tr}(A)>2,\quad \text{tr}(B)>2,\quad \text{tr}(ABA^{-1}B^{-1}) = 2$$ Prove that $A$ and $B$ have a common eigenvector.

2025 Romania National Olympiad, 4

Tags: trace , matrix , linear algebra

Let $A,B \in \mathcal{M}_n(\mathbb{C})$ be two matrices such that $A+B=AB+BA$. Prove that: a) if $n$ is odd, then $\det(AB-BA)=0$; b) if $\text{tr}(A)\neq \text{tr}(B)$, then $\det(AB-BA)=0$.

2022 OMpD, 2

Tags: algebra , linear algebra , matrix , matrix algebra , trace

Let $p \geq 3$ be a prime number and let $A$ be a matrix of order $p$ with complex entries. Assume that $\text{Tr}(A) = 0$ and $\det(A - I_p) \neq 0$. Prove that $A^p \neq I_p$. Note: $\text{Tr}(A)$ is the sum of the main diagonal elements of $A$ and $I_p$ is the identity matrix of order $p$.

2024 Brazil Undergrad MO, 5

Tags: matrix , linear algebra , trace

Let $ A $ be a $ 2 \times 2 $ matrix with integer entries and $\det A \neq 0$. If the sequence $\operatorname{tr}(A^n)$, for $ n = 1, 2, 3, \ldots $, is bounded, show that \[ A^{12} = I \quad \text{or} \quad (A^2 - I)^2 = O. \] Here, $ I $ and $ O $ denote the identity and zero matrices, respectively, and $\operatorname{tr}$ denotes the trace of the matrix (the sum of the elements on the main diagonal).

2016 Korea USCM, 3

Tags: matrix , linear algebra , trace , invertible

Given positive integers $m,n$ and a $m\times n$ matrix $A$ with real entries. (1) Show that matrices $X = I_m + AA^T$ and $Y = I_n + A^T A$ are invertible. ($I_l$ is the $l\times l$ unit matrix.) (2) Evaluate the value of $\text{tr}(X^{-1}) - \text{tr}(Y^{-1})$.

2018 Korea USCM, 2

Tags: linear algebra , rank , trace , matrix

Suppose a $n\times n$ real matrix $A$ satisfies $\text{tr}(A)=2018$, $\text{rank}(A)=1$. Prove that $A^2=2018 A$.

2025 VJIMC, 4

Tags: linear algebra , matrix , algebra , polynomial , trace , minimal polynomial

Let $A$ be an $n\times n$ real matrix with minimal polynomial $x^n + x - 1$. Prove that the trace of $(nA^{n-1} + I)^{-1}A^{n-2}$ is zero.