Olympiad problems

2021 Brazil Undergrad MO, Problem 1

Consider the matrices like $$M= \left( \begin{array}{ccc} a & b & c \\ c & a & b \\ b & c & a \end{array} \right)$$ such that $det(M) = 1$. Show that a) There are infinitely many matrices like above with $a,b,c \in \mathbb{Q}$ b) There are finitely many matrices like above with $a,b,c \in \mathbb{Z}$

2020 Brazil Undergrad MO, Problem 4

Tags: linear algebra , college contest , matrix algebra

For each of the following, provide proof or a counterexample: a) Every $2\times2$ matrix with real entries can we written as the sum of the squares of two $2\times2$ matrices with real entries. b) Every $3\times3$ matrix with real entries can we written as the sum of the squares of two $3\times3$ matrices with real entries.

2023 Brazil Undergrad MO, 4

Tags: matrix , matrix algebra , linear algebra , eigenvalue and eigenvector

Let $M_2(\mathbb{Z})$ be the set of $2 \times 2$ matrices with integer entries. Let $A \in M_2(\mathbb{Z})$ such that $$A^2+5I=0,$$ where $I \in M_2(\mathbb{Z})$ and $0 \in M_2(\mathbb{Z})$ denote the identity and null matrices, respectively. Prove that there exists an invertible matrix $C \in M_2(\mathbb{Z})$ with $C^{-1} \in M_2(\mathbb{Z})$ such that $$CAC^{-1} = \begin{pmatrix} 1 & 2\\ -3 & -1 \end{pmatrix} \text{ ou } CAC^{-1} = \begin{pmatrix} 0 & 1\\ -5 & 0 \end{pmatrix}.$$

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 $p$.