Found problems: 48
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 $ ).
2004 Nicolae Coculescu, 3
Solve in $ \mathcal{M}_2(\mathbb{R}) $ the equation $ X^3+X+2I=0. $
[i]Florian Dumitrel[/i]
2006 Cezar Ivănescu, 2
Prove that the set $ \left\{ \left. \begin{pmatrix} \frac{1-2x^3}{3x^2} & \frac{1+x^3}{3x^2} & \frac{1+x^3}{3x^2} \\ \frac{1+x^3}{3x^2} & \frac{1-2x^3}{3x^2} & \frac{1+x^3}{3x^2} \\ \frac{1+x^3}{3x^2} & \frac{1+x^3}{3x^2} & \frac{1-2x^3}{3x^2}\end{pmatrix}\right| x\in\mathbb{R}^{*} \right\} $ along with the usual multiplication of matrices form a group, determine an isomorphism between this group and the group of multiplicative real numbers.
2024 Mexican University Math Olympiad, 2
Let \( A \) and \( B \) be two square matrices with complex entries such that \( A + B = AB \), \( A = A^* \), and \( A \) has all distinct eigenvalues. Prove that there exists a polynomial \( P \) with complex coefficients such that \( P(A) = B \).
2004 Spain Mathematical Olympiad, Problem 1
We have a set of ${221}$ real numbers whose sum is ${110721}$. It is deemed that the numbers form a rectangular table such that every row as well as the first and last columns are arithmetic progressions of more than one element. Prove that the sum of the elements in the four corners is equal to ${2004}$.
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?
2015 VJIMC, 1
[b]Problem 1 [/b]
Let $A$ and $B$ be two $3 \times 3$ matrices with real entries. Prove that
$$ A-(A^{-1} +(B^{-1}-A)^{-1})^{-1} =ABA\ , $$
provided all the inverses appearing on the left-hand side of the equality exist.
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. $
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?
2024 Mexican University Math Olympiad, 4
Given \( b > 0 \), consider the following matrix:
\[
B = \begin{pmatrix} b & b^2 \\ b^2 & b^3 \end{pmatrix}
\]
Denote by \( e_i \) the top left entry of \( B^i \). Prove that the following limit exists and calculate its value:
\[
\lim_{i \to \infty} \sqrt[i]{e_i}.
\]
2019 District Olympiad, 3
Let $n$ be an odd natural number and $A,B \in \mathcal{M}_n(\mathbb{C})$ be two matrices such that $(A-B)^2=O_n.$ Prove that $\det(AB-BA)=0.$
2019 Korea USCM, 8
$M_n(\mathbb{C})$ is the vector space of all complex $n\times n$ matrices. Given a linear map $T:M_n(\mathbb{C})\to M_n(\mathbb{C})$ s.t. $\det (A)=\det(T(A))$ for every $A\in M_n(\mathbb{C})$.
(1) If $T(A)$ is the zero matrix, then show that $A$ is also the zero matrix.
(2) Prove that $\text{rank} (A)=\text{rank} (T(A))$ for any $A\in M_n(\mathbb{C})$.
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
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
$$ \Delta_n (x)=\begin{vmatrix}1 & 1-m & 1-m & \cdots & 1-m & 1-m & -m \\ -1 & \binom{m}{x} & 0 & \cdots & 0 & 0 & 0 \\ 0 & -1 & \binom{m}{x} & \cdots & 0 & 0 & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & \cdots & -1 & \binom{m}{x} & 0 \\ 0 & 0 & 0 & \cdots & 0 & -1 & \binom{m}{x}\end{vmatrix} . $$
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]
2022 SEEMOUS, 3
Let $\alpha \in \mathbb{C}\setminus \{0\}$ and $A \in \mathcal{M}_n(\mathbb{C})$, $A \neq O_n$, be such that
$$A^2 + (A^*)^2 = \alpha A\cdot A^*,$$
where $A^* = (\bar A)^T.$ Prove that $\alpha \in \mathbb{R}$, $|\alpha| \le 2$. and $A\cdot A^* = A^*\cdot A.$
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]
2017 Simon Marais Mathematical Competition, A3
For each positive integer $n$, let $M(n)$ be the $n\times n$ matrix whose $(i,j)$ entry is equal to $1$ if $i+1$ is divisible by $j$, and equal to $0$ otherwise. Prove that $M(n)$ is invertible if and only if $n+1$ is square-free. (An integer is [i]square-free[/i] if it is not divisible by a square of an integer larger than $1$.)
2024 IMC, 7
Let $n$ be a positive integer. Suppose that $A$ and $B$ are invertible $n \times n$ matrices with complex entries such that $A+B=I$ (where $I$ is the identity matrix) and
\[(A^2+B^2)(A^4+B^4)=A^5+B^5.\]
Find all possible values of $\det(AB)$ for the given $n$.
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:
$$ \det\left( (AB)^m-(BA)^m\right)\cdot\det\left( (AB)^n-(BA)^n\right)\ge 0. $$
2021 Alibaba Global Math Competition, 3
Given positive integers $k \ge 2$ and $m$ sufficiently large. Let $\mathcal{F}_m$ be the infinite family of all the (not necessarily square) binary matrices which contain exactly $m$ 1's. Denote by $f(m)$ the maximum integer $L$ such that for every matrix $A \in \mathcal{F}_m$, there always exists a binary matrix $B$ of the same dimension such that (1) $B$ has at least $L$ 1-entries; (2) every entry of $B$ is less or equal to the corresponding entry of $A$; (3) $B$ does not contain any $k \times k$ all-1 submatrix. Show the equality
\[\lim_{m \to \infty} \frac{\ln f(m)}{\ln m}=\frac{k}{k+1}.\]
2022 Romania National Olympiad, P4
Let $A,B\in\mathcal{M}_n(\mathbb{C})$ such that $A^2+B^2=2AB.$ Prove that for any complex number $x$\[\det(A-xI_n)=\det(B-xI_n).\][i]Mihai Opincariu and Vasile Pop[/i]
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 $\det A$.
2013 Bogdan Stan, 3
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]
1996 Romania National Olympiad, 4
Let $A,B,C,D \in \mathcal{M}_n(\mathbb{C}),$ $A$ and $C$ invertible. Prove that if $A^k B = C^k D$ for any positive integer $k,$ then $B=D.$