Olympiad problems

2017 Miklós Schweitzer, 3

For every algebraic integer $\alpha$ define its positive degree $\text{deg}^+(\alpha)$ to be the minimal $k\in\mathbb{N}$ for which there exists a $k\times k$ matrix with non-negative integer entries with eigenvalue $\alpha$. Prove that for any $n\in\mathbb{N}$, every algebraic integer $\alpha$ with degree $n$ satisfies $\text{deg}^+(\alpha)\le 2n$.

2011 SEEMOUS, Problem 2

Tags: eigenvalue , matrix , linear algebra

Let $A=(a_{ij})$ be a real $n\times n$ matrix such that $A^n\ne0$ and $a_{ij}a_{ji}\le0$ for all $i,j$. Prove that there exist two nonreal numbers among eigenvalues of $A$.

2022 SEEMOUS, 1

Tags: linear algebra , matrices , sylvester inequality , eigenvalue

Let $A, B \in \mathcal{M}_n(\mathbb{C})$ be such that $AB^2A = AB$. Prove that: a) $(AB)^2 = AB.$ b) $(AB - BA)^3 = O_n.$

2006 Cezar Ivănescu, 2

Tags: linear algebra , eigenvalue , complex number , group theory , matrix

[b]a)[/b] Let $ a,b,c $ be three complex numbers. Prove that the element $ \begin{pmatrix} a & a-b & a-b \\ 0 & b & b-c \\ 0 & 0 & c \end{pmatrix} $ has finite order in the multiplicative group of $ 3\times 3 $ complex matrices if and only if $ a,b,c $ have finite orders in the multiplicative group of complex numbers. [b]b)[/b] Prove that a $ 3\times 3 $ real matrix $ M $ has positive determinant if there exists a real number $ \lambda\in\left( 0,\sqrt[3]{4} \right) $ such that $ A^3=\lambda A+I. $ [i]Cristinel Mortici[/i]

2013 Bogdan Stan, 3

Tags: linear algebra , eigenvalue , matries , matrices

Let be four $ n\times n $ real matrices $ A,B,C,D $ having the property that $ C+D\sqrt{-1} $ is the inverse of $ A+B\sqrt{-1} . $ Show that $ \left| \det\left( A+B\sqrt{-1} \right) \right|^2\cdot\left| \det C \right| =\det A. $ [i]Vasile Pop[/i]

2021 Brazil Undergrad MO, Problem 5

Tags: linear algebra , matrix , eigenvalue

Find all triplets $(\lambda_1,\lambda_2,\lambda_3) \in \mathbb{R}^3$ such that there exists a matrix $A_{3 \times 3}$ with all entries being non-negative reals whose eigenvalues are $\lambda_1,\lambda_2,\lambda_3$.

2016 VJIMC, 3

Tags: linear algebra , matrix , eigenvalue

For $n \geq 3$ find the eigenvalues (with their multiplicities) of the $n \times n$ matrix $$\begin{bmatrix} 1 & 0 & 1 & 0 & 0 & 0 & \dots & \dots & 0 & 0\\ 0 & 2 & 0 & 1 & 0 & 0 & \dots & \dots & 0 & 0\\ 1 & 0 & 2 & 0 & 1 & 0 & \dots & \dots & 0 & 0\\ 0 & 1 & 0 & 2 & 0 & 1 & \dots & \dots & 0 & 0\\ 0 & 0 & 1 & 0 & 2 & 0 & \dots & \dots & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 2 & \dots & \dots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & & \vdots & \vdots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 0 & 1 \end{bmatrix}$$

2022 VTRMC, 5

Tags: linear algebra , matrix , eigenvalue

Let $A$ be an invertible $n \times n$ matrix with complex entries. Suppose that for each positive integer $m$, there exists a positive integer $k_m$ and an $n \times n$ invertible matrix $B_m$ such that $A^{k_m m} = B_m A B_m ^{-1}$. Show that all eigenvalues of $A$ are equal to $1$.

2018 Korea USCM, 7

Tags: matrix , linear algebra , positive definite , vector , eigenvalue

Suppose a $3\times 3$ matrix $A$ satisfies $\mathbf{v}^t A \mathbf{v} > 0$ for any vector $\mathbf{v} \in\mathbb{R}^3 -\{0\}$. (Note that $A$ may not be a symmetric matrix.) (1) Prove that $\det(A)>0$. (2) Consider diagonal matrix $D=\text{diag}(-1,1,1)$. Prove that there's exactly one negative real among eigenvalues of $AD$.