Olympiad problems

2023 District Olympiad, P4

Let $A{}$ and $B{}$ be $3\times 3{}$ matrices with complex entries, satisfying $A^2=B^2=O_3$. Prove that if $A{}$ and $B{}$ commute, then $AB=O_3$. Is the converse true?

2018 Romania National Olympiad, 4

Tags: rank , matrices , linear algebra , matrix

Let $n$ be an integer with $n \geq 2$ and let $A \in \mathcal{M}_n(\mathbb{C})$ such that $\operatorname{rank} A \neq \operatorname{rank} A^2.$ Prove that there exists a nonzero matrix $B \in \mathcal{M}_n(\mathbb{C})$ such that $$AB=BA=B^2=0$$ [i]Cornel Delasava[/i]

2023 Romania National Olympiad, 3

Tags: nilpotent , rank , linear algebra

Let $n$ be a natural number $n \geq 2$ and matrices $A,B \in M_{n}(\mathbb{C}),$ with property $A^2 B = A.$ a) Prove that $(AB - BA)^2 = O_{n}.$ b) Show that for all natural number $k$, $k \leq \frac{n}{2}$ there exist matrices $A,B \in M_{n}(\mathbb{C})$ with property stated in the problem such that $rank(AB - BA) = k.$

2022 IMC, 7

Tags: linear algebra , idempotence , rank

Let $A_1, \ldots, A_k$ be $n\times n$ idempotent complex matrices such that $A_iA_j = -A_iA_j$ for all $1 \leq i < j \leq k$. Prove that at least one of the matrices has rank not exceeding $\frac{n}{k}$.

2022 District Olympiad, P4

Tags: matrix , rank

Let $A\in\mathcal{M}_n(\mathbb{C})$ where $n\geq 2.$ Prove that if $m=|\{\text{rank}(A^k)-\text{rank}(A^{k+1})":k\in\mathbb{N}^*\}|$ then $n+1\geq m(m+1)/2.$

1981 Putnam, B4

Tags: matrix , rank , rank of matrix

Let $V$ be a set of $5\times7$ matrices, with real entries and closed under addition and scalar multiplication. Prove or disprove the following assertion: If $V$ contains matrices of ranks $0, 1, 2, 4,$ and $5$, then it also contains a matrix of rank $3$.

2019 Korea USCM, 1

Tags: linear algebra , too easy , rank , matrix

$A = \begin{pmatrix} 2019 & 2020 & 2021 \\ 2020 & 2021 & 2022 \\ 2021 & 2022 & 2023 \end{pmatrix}$. Find $\text{rank}(A)$.

2016 Romania National Olympiad, 2

Tags: linear algebra , rank

Consider a natural number, $ n\ge 2, $ and three $ n\times n $ complex matrices $ A,B,C $ such that $ A $ is invertible, $ B $ is formed by replacing the first line of $ A $ with zeroes, and $ C $ is formed by putting the last $ n-1 $ lines of $ A $ above a line of zeroes. Prove that: [b]a)[/b] $ \text{rank} \left( A^{-1} B \right) = \text{rank} \left( \left( A^{-1} B\right)^2 \right) =\cdots =\text{rank} \left( \left( A^{-1} B\right)^n \right) $ [b]b)[/b] $ \text{rank} \left( A^{-1} C \right) > \text{rank} \left( \left( A^{-1} C\right)^2 \right) >\cdots >\text{rank} \left( \left( A^{-1} C\right)^n \right) $