Found problems: 48
2018 Romania National Olympiad, 4
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]
2001 Miklós Schweitzer, 3
How many minimal left ideals does the full matrix ring $M_n(K)$ of $n\times n$ matrices over a field $K$ have?
2016 IMC, 2
Let $k$ and $n$ be positive integers. A sequence $\left( A_1, \dots , A_k \right)$ of $n\times n$ real matrices is [i]preferred[/i] by Ivan the Confessor if $A_i^2\neq 0$ for $1\le i\le k$, but $A_iA_j=0$ for $1\le i$, $j\le k$ with $i\neq j$. Show that $k\le n$ in all preferred sequences, and give an example of a preferred sequence with $k=n$ for each $n$.
(Proposed by Fedor Petrov, St. Petersburg State University)
1987 Traian Lălescu, 1.2
Let be a natural number $ n, $ a complex number $ a, $ and two matrices $ \left( a_{pq}\right)_{1\le q\le n}^{1\le p\le n} ,\left( b_{pq}\right)_{1\le q\le n}^{1\le p\le n}\in\mathcal{M}_n(\mathbb{C} ) $ such that
$$ b_{pq} =a^{p-q}\cdot a_{pq},\quad\forall p,q\in\{ 1,2,\ldots ,n\} . $$
Calculate the determinant of $ B $ (in function of $ a $ and the determinant of $ A $ ).
1976 Spain Mathematical Olympiad, 6
Given a square matrix $M$ of order $n$ over the field of numbers real, find, as a function of $M$, two matrices, one symmetric and one antisymmetric, such that their sum is precisely $ M$.
2024 OMpD, 2
Let \( n \) be a positive integer, and let \( A \) and \( B \) be \( n \times n \) matrices with real coefficients such that
\[
ABBA - BAAB = A - B.
\]
(a) Prove that \( \text{Tr}(A) = \text{Tr}(B) \) and that \( \text{Tr}(A^2) = \text{Tr}(B^2) \).
(b) If \(BA^2B= A^2B^2\) and \(AB^2A= B^2A^2\), prove that \( \det A = \det B \).
Note: \( \text{Tr}(X) \) denotes the trace of \( X \), which is the sum of the elements on its main diagonal, and \( \det X \) denotes the determinant of \( X \).
2015 District Olympiad, 2
Let be two matrices $ A,B\in M_2\left(\mathbb{R}\right) $ that satisfy the equality $ \left( A-B\right)^2 =O_2. $
[b]a)[/b] Show that $ \det\left( A^2-B^2\right) =\left( \det A -\det B\right)^2. $
[b]b)[/b] Demonstrate that $ \det\left( AB-BA\right) =0\iff \det A=\det B. $
2019 LIMIT Category C, Problem 10
Let $A\in M_3(\mathbb Z)$ such that $\det(A)=1$. What is the maximum possible number of entries of $A$ that are even?
2022 Romania National Olympiad, P2
Let $\mathcal{F}$ be the set of pairs of matrices $(A,B)\in\mathcal{M}_2(\mathbb{Z})\times\mathcal{M}_2(\mathbb{Z})$ for which there exists some positive integer $k$ and matrices $C_1,C_2,\ldots, C_k\in\{A,B\}$ such that $C_1C_2\cdots C_k=O_2.$ For each $(A,B)\in\mathcal{F},$ let $k(A,B)$ denote the minimal positive integer $k$ which satisfies the latter property.
[list=a]
[*]Let $(A,B)\in\mathcal{F}$ with $\det(A)=0,\det(B)\neq 0$ and $k(A,B)=p+2$ for some $p\in\mathbb{N}^*.$ Show that $AB^pA=O_2.$
[*]Prove that for any $k\geq 3$ there exists a pair $(A,B)\in\mathcal{F}$ such that $k(A,B)=k.$
[/list][i]Bogdan Blaga[/i]
2024 VJIMC, 2
Here is a problem we (me and my colleagues) suggested and was given at the competition this year. The problem statement is very natural and short. However, we have not seen such a problem before.
A real $2024 \times 2024$ matrix $A$ is called nice if $(Av, v) = 1$ for every vector $v\in \mathbb{R}^{2024}$ with unit norm.
a) Prove that the only nice matrix such that all of its eigenvalues are real is the identity matrix.
b) Find an example of a nice non-identity matrix
1986 Traian Lălescu, 2.1
Show that for any natural numbers $ m,n\ge 3, $ the equation $ \Delta_n (x)=0 $ has exactly two distinct solutions, where
$$ \Delta_n (x)=\begin{vmatrix}1 & 1-m & 1-m & \cdots & 1-m & 1-m & -m \\ -1 & \binom{m}{x} & 0 & \cdots & 0 & 0 & 0 \\ 0 & -1 & \binom{m}{x} & \cdots & 0 & 0 & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & \cdots & -1 & \binom{m}{x} & 0 \\ 0 & 0 & 0 & \cdots & 0 & -1 & \binom{m}{x}\end{vmatrix} . $$
2015 IMC, 1
For any integer $n\ge 2$ and two $n\times n$ matrices with real
entries $A,\; B$ that satisfy the equation
$$A^{-1}+B^{-1}=(A+B)^{-1}\;$$
prove that $\det (A)=\det(B)$.
Does the same conclusion follow for matrices with complex entries?
(Proposed by Zbigniew Skoczylas, Wroclaw University of Technology)
2014 CHMMC (Fall), 2
A matrix $\begin{bmatrix}
x & y \\
z & w
\end{bmatrix}$ has square root $\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}$ if
$$\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}^2
=
\begin{bmatrix}
a^2 + bc &ab + bd \\
ac + cd & bc + d^2
\end{bmatrix}
=
\begin{bmatrix}
x & y \\
z & w
\end{bmatrix}$$
Determine how many square roots the matrix $\begin{bmatrix}
2 & 2 \\
3 & 4
\end{bmatrix}$ has (complex coefficients are allowed).
2023 Miklós Schweitzer, 10
Let $n\geqslant2$ be a natural number. Show that there is no real number $c{}$ for which \[\exp\left(\frac{T+S}{2}\right)\leqslant c\cdot \frac{\exp(T)+\exp(S)}{2}\]is satisfied for any self-adjoint $n\times n$ complex matrices $T{}$ and $S{}$. (If $A{}$ and $B{}$ are self-adjoint $n\times n$ matrices, $A\leqslant B$ means that $B-A$ is positive semi-definite.)
1972 Spain Mathematical Olympiad, 1
Let $K$ be a ring with unit and $M$ the set of $2 \times 2$ matrices constituted with elements of $K$. An addition and a multiplication are defined in $M$ in the usual way between arrays. It is requested to:
a) Check that $M$ is a ring with unit and not commutative with respect to the laws of defined composition.
b) Check that if $K$ is a commutative field, the elements of$ M$ that have inverse they are characterized by the condition $ad - bc \ne 0$.
c) Prove that the subset of $M$ formed by the elements that have inverse is a multiplicative group.
2015 District Olympiad, 3
Find all natural numbers $ k\ge 1 $ and $ n\ge 2, $ which have the property that there exist two matrices $ A,B\in M_n\left(\mathbb{Z}\right) $ such that $ A^3=O_n $ and $ A^kB +BA=I_n. $
2022 SEEMOUS, 1
Let $A, B \in \mathcal{M}_n(\mathbb{C})$ be such that $AB^2A = AB$. Prove that:
a) $(AB)^2 = AB.$
b) $(AB - BA)^3 = O_n.$
2014 VTRMC, Problem 6
Let $S$ denote the set of $2$ by $2$ matrices with integer entries and determinant $1$, and let $T$ denote those matrices of $S$ which are congruent to the identity matrix $I\pmod3$ (so $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in T$ means that $a,b,c,d\in\mathbb Z,ad-bc=1,$ and $3$ divides $b,c,a-1,d-1$).
(a) Let $f:T\to\mathbb R$ be a function such that for every $X,Y\in T$ with $Y\ne I$, either $f(XY)>f(X)$ or $f(XY^{-1})>f(X)$. Show that given two finite nonempty subsets $A,B$ of $T$, there are matrices $a\in A$ and $b\in B$ such that if $a'\in A$, $b'\in B$ and $a'b'=ab$, then $a'=a$ and $b'=b$.
(b) Show that there is no $f:S\to\mathbb R$ such that for every $X,Y\in S$ with $Y\ne\pm I$, either $f(XY)>f(X)$ or $f(XY^{-1})>f(X)$.
1985 Traian Lălescu, 1.3
Let be two matrices $ A,B\in M_2\left(\mathbb{R}\right) $ and two natural numbers $ m,n. $ Prove that:
$$ \det\left( (AB)^m-(BA)^m\right)\cdot\det\left( (AB)^n-(BA)^n\right)\ge 0. $$
2015 IMC, 9
An $n \times n$ complex matrix $A$ is called \emph{t-normal} if
$AA^t = A^t A$ where $A^t$ is the transpose of $A$. For each $n$,
determine the maximum dimension of a linear space of complex $n
\times n$ matrices consisting of t-normal matrices.
Proposed by Shachar Carmeli, Weizmann Institute of Science
2016 District Olympiad, 1
Let $ A\in M_2\left( \mathbb{C}\right) $ such that $ \det \left( A^2+A+I_2\right) =\det \left( A^2-A+I_2\right) =3. $
Prove that $ A^2\left( A^2+I_2\right) =2I_2. $
2003 Romania National Olympiad, 2
Let be eight real numbers $ 1\le a_1< a_2< a_3< a_4,x_1<x_2<x_3<x_4. $ Prove that
$$ \begin{vmatrix}a_1^{x_1} & a_1^{x_2} & a_1^{x_3} & a_1^{x_4} \\
a_2^{x_1} & a_2^{x_2} & a_2^{x_3} & a_2^{x_4} \\
a_3^{x_1} & a_3^{x_2} & a_3^{x_3} & a_3^{x_4} \\
a_4^{x_1} & a_4^{x_2} & a_4^{x_3} & a_4^{x_4} \\
\end{vmatrix} >0. $$
[i]Marian Andronache, Ion Savu[/i]
ICMC 4, 2
Let \(A\) be a square matrix with entries in the field \(\mathbb Z / p \mathbb Z\) such that \(A^n - I\) is invertible for every positive integer \(n\). Prove that there exists a positive integer \(m\) such that \(A^m = 0\).
[i](A matrix having entries in the field \(\mathbb Z / p \mathbb Z\) means that two matrices are considered the same if each pair of corresponding entries differ by a multiple of \(p\).)[/i]
[i]Proposed by Tony Wang[/i]
2000 Moldova National Olympiad, Problem 7
Prove that for any positive integer $n$ there exists a matrix of the form
$$A=\begin{pmatrix}1&a&b&c\\0&1&a&b\\0&0&1&a\\0&0&0&1\end{pmatrix},$$
(a) with nonzero entries,
(b) with positive entries,
such that the entries of $A^n$ are all perfect squares.
2015 VTRMC, Problem 3
Let $(a_i)_{1\le i\le2015}$ be a sequence consisting of $2015$ integers, and let $(k_i)_{1\le i\le2015}$ be a sequence of $2015$ positive integers (positive integer excludes $0$). Let
$$A=\begin{pmatrix}a_1^{k_1}&a_1^{k_2}&\cdots&a_1^{k_{2015}}\\a_2^{k_1}&a_2^{k_2}&\cdots&a_2^{k_{2015}}\\\vdots&\vdots&\ddots&\vdots\\a_{2015}^{k_1}&a_{2015}^{k_2}&\cdots&a_{2015}^{k_{2015}}\end{pmatrix}.$$Prove that $2015!$ divides $\det A$.