Found problems: 48
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 \( X \).
2019 Brazil Undergrad MO, 1
Let $ I $ and $ 0 $ be the square identity and null matrices, both of size $ 2019 $. There is a square matrix $A$
with rational entries and size $ 2019 $ such that:
a) $ A ^ 3 + 6A ^ 2-2I = 0 $?
b) $ A ^ 4 + 6A ^ 3-2I = 0 $?
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.
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$.
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.)
2018 District Olympiad, 1
Show that if $n\ge 2$ is an integer, then there exist invertible matrices $A_1, A_2, \ldots, A_n \in \mathcal{M}_2(\mathbb{R})$ with non-zero entries such that:
\[A_1^{-1} + A_2^{-1} + \ldots + A_n^{-1} = (A_1 + A_2 + \ldots + A_n)^{-1}.\]
[hide=Edit.]
The $77777^{\text{th}}$ topic in College Math :coolspeak:
[/hide]
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 $A\in M_n(\mathbb{C})$.
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).
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
\[\lim_{m \to \infty} \frac{\ln f(m)}{\ln m}=\frac{k}{k+1}.\]
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. $
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]
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 \( P(A) = B \).
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]
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?
2004 Nicolae Coculescu, 3
Solve in $ \mathcal{M}_2(\mathbb{R}) $ the equation $ X^3+X+2I=0. $
[i]Florian Dumitrel[/i]
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
$$ \Delta_n (x)=\begin{vmatrix}1 & 1-m & 1-m & \cdots & 1-m & 1-m & -m \\ -1 & \binom{m}{x} & 0 & \cdots & 0 & 0 & 0 \\ 0 & -1 & \binom{m}{x} & \cdots & 0 & 0 & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & \cdots & -1 & \binom{m}{x} & 0 \\ 0 & 0 & 0 & \cdots & 0 & -1 & \binom{m}{x}\end{vmatrix} . $$
1967 Putnam, A2
Define $S_0$ to be $1.$ For $n \geq 1 $, let $S_n $ be the number of $n\times n $ matrices whose elements are nonnegative integers with the property that $a_{ij}=a_{ji}$ (for $i,j=1,2,\ldots, n$) and where $\sum_{i=1}^{n} a_{ij}=1$ (for $j=1,2,\ldots, n$). Prove that
a) $S_{n+1}=S_{n} +nS_{n-1}.$
b) $\sum_{n=0}^{\infty} S_{n} \frac{x^{n}}{n!} =\exp \left(x+\frac{x^{2}}{2}\right).$
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]
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. $
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:
\[
\lim_{i \to \infty} \sqrt[i]{e_i}.
\]
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 $A\cdot A^* = A^*\cdot A.$
2019 LIMIT Category C, Problem 7
Let $O(4,\mathbb Z)$ be the set of all $4\times4$ orthogonal matrices over $\mathbb Z$, i.e., $A^tA=I=AA^t$. Then $|O(4,\mathbb Z)|$ is
1996 Romania National Olympiad, 3
Let $A, B \in M_2(\mathbb{R})$ such as $det(AB+BA)\leq 0$. Prove that $$det(A^2+B^2)\geq 0$$
2012 Mathcenter Contest + Longlist, 1 sl8
For matrices $A=[a_{ij}]_{m \times m}$ and $B=[b_{ij}]_{m \times m}$ where $A,B \in \mathbb{Z} ^{m \times m}$ let $A \equiv B \pmod{n}$ only if $a_{ij} \equiv b_{ij} \pmod{n}$ for every $i,j \in \{ 1,2,...,m \}$, that's $A-B=nZ$ for some $Z \in \mathbb{Z}^{m \times m}$. (The symbol $A \in \mathbb{Z} ^{m \times m}$ means that every element in $A$ is an integer.)
Prove that for $A \in \mathbb{Z} ^{m \times m}$ there is $B \in \mathbb{Z} ^{m \times m}$ , where $AB \equiv I \pmod{n }$ only if $(\det (A),n)=1$ and find the value of $B$ in the form of $A$ where $I$ represents the dimensional identity matrix $m \times m$.
[i](PP-nine)[/i]
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]