Found problems: 638
Let be a $ 2\times 2 $ real matrix $ A $ that has the property that $ \left| A^d-I_2 \right| =\left| A^d+I_2 \right| , $ for all $ d\in\{ 2014,2016 \} . $
Prove that $ \left| A^n-I_2 \right| =\left| A^n+I_2 \right| , $ for any natural number $ n. $
Let $A$ be an $n\times n$ complex matrix such that $A \ne \lambda I_{n}$ for all $\lambda \in \mathbb{C}$. Prove that $A$ is similar to a matrix having at most one non-zero entry on the maindiagonal.
Let $A,B\in \mathcal{M}_n(\mathbb{C})$ be two $n \times n$ matrices such that
\[ A^2B+BA^2=2ABA \]
Prove there exists $k\in \mathbb{N}$ such that
\[ (AB-BA)^k=\mathbf{0}_n\]
Here $\mathbf{0}_n$ is the null matrix of order $n$.
Let $x_1,\ldots, x_n$ and $y_1,\ldots, y_n$ be real numbers. Let $A = (a_{ij})_{1\leq i,j\leq n}$ be the matrix with entries \[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\] Suppose that $B$ is an $n\times n$ matrix with entries $0$, $1$ such that the sum of the elements in each row and each column of $B$ is equal to the corresponding sum for the matrix $A$. Prove that $A=B$.
Given a finite tree $ T$ and isomorphism $ f: T\rightarrow T$. Prove that either there exist a vertex $ a$ such that $ f(a)\equal{}a$ or there exist two neighbor vertices $ a$, $ b$ such that $ f(a)\equal{}b$, $ f(b)\equal{}a$.
Let $ n(\ge2) $ be a positive integer. Find the minimum $ m $, so that there exists $x_{ij}(1\le i ,j\le n)$ satisfying:
(1)For every $1\le i ,j\le n, x_{ij}=max\{x_{i1},x_{i2},...,x_{ij}\} $ or $ x_{ij}=max\{x_{1j},x_{2j},...,x_{ij}\}.$
(2)For every $1\le i \le n$, there are at most $m$ indices $k$ with $x_{ik}=max\{x_{i1},x_{i2},...,x_{ik}\}.$
(3)For every $1\le j \le n$, there are at most $m$ indices $k$ with $x_{kj}=max\{x_{1j},x_{2j},...,x_{kj}\}.$
Let
\begin{align*}X&=\begin{pmatrix}7&8&9\\8&-9&-7\\-7&-7&9\end{pmatrix}\\Y&=\begin{pmatrix}9&8&-9\\8&-7&7\\7&9&8\end{pmatrix}.\end{align*}Let $A=Y^{-1}X$ and let $B$ be the inverse of $X^{-1}+A^{-1}$. Find a matrix $M$ such that $M^2=XY-BY$ (you may assume that $A$ and $X^{-1}+A^{-1}$ are invertible).
Let $ n\ge 2$ be an integer. What is the minimal and maximal possible rank of an $ n\times n$ matrix whose $ n^{2}$ entries are precisely the numbers $ 1, 2, \ldots, n^{2}$?
Let $\{x\}$ be a sequence of positive reals $x_1, x_2, \ldots, x_n$, defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$. And for $n \geq 1$ we have:
\[x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}.\]
Show that this sequence has a finite limit. Determine this limit.
A country has $n$ cities, labelled $1,2,3,\dots,n$. It wants to build exactly $n-1$ roads between certain pairs of cities so that every city is reachable from every other city via some sequence of roads. However, it is not permitted to put roads between pairs of cities that have labels differing by exactly $1$, and it is also not permitted to put a road between cities $1$ and $n$. Let $T_n$ be the total number of possible ways to build these roads.
(a) For all odd $n$, prove that $T_n$ is divisible by $n$.
(b) For all even $n$, prove that $T_n$ is divisible by $n/2$.
Let $A\in\mathcal{M}_n(\mathbb{C})$ be an antisymmetric matrix, i.e. $A=-A^t.$[list=a]
[*]Prove that if $A\in\mathcal{M}_n(\mathbb{R})$ and $A^2=O_n$ then $A=O_n.$
[*]Assume that $n{}$ is odd. Prove that if $A{}$ is the adjoint of another matrix $B\in\mathcal{M}_n(\mathbb{C})$ then $A^2=O_n.$
[/list]
Let $n\in \mathbb{N}^*$ and a matrix $A\in \mathcal{M}_n(\mathbb{C}),\ A=(a_{ij})_{1\le i, j\le n}$ such that:
\[a_{ij}+a_{jk}+a_{ki}=0,\ (\forall)i,j,k\in \{1,2,\ldots,n\}\]
Prove that $\text{rank}\ A\le 2$.
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 $3$.
Let $\mathcal{M}$ be the set of all $2021 \times 2021$ matrices with at most two entries in each row equal to $1$ and all other entries equal to $0$.
Determine the size of the set $\{ \det A : A \in M \}$.
[i]Here $\det A$ denotes the determinant of the matrix $A$.[/i]
If $ P(x),Q(x),R(x)$, and $ S(x)$ are all polynomials such that \[ P(x^5)\plus{}xQ(x^5)\plus{}x^2R(x^5)\equal{}(x^4\plus{}x^3\plus{}x^2\plus{}x\plus{}1)S(x),\] prove that $ x\minus{}1$ is a factor of $ P(x)$.
Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$, $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.
Let $\{k_1, k_2, \dots , k_m\}$ a set of $m$ integers. Show that there exists a matrix $m \times m$ with integers entries $A$ such that each of the matrices $A + k_jI, 1 \leq j \leq m$ are invertible and their entries have integer entries (here $I$ denotes the identity matrix).
Solve in $(\mathbb{R}_{+}^{*})^{4}$ the following system :
$\left\{\begin{matrix}
x+y+z+t=4\\
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{t}=5-\frac{1}{xyzt}
\end{matrix}\right.$
Let $n$ be a positive integer and let $A$, $B$ be two complex nonsingular $n \times n$ matrices such that
\[A^2B-2ABA+BA^2=0.\]
Prove that the matrix $AB^{-1}A^{-1}B-I_n$ is nilpotent.
Let $a,b,c$ be 3 integers, $b$ odd, and define the sequence $\{x_n\}_{n\geq 0}$ by $x_0=4$, $x_1=0$, $x_2=2c$, $x_3=3b$ and for all positive integers $n$ we have
\[ x_{n+3} = ax_{n-1}+bx_n + cx_{n+1} . \]
Prove that for all positive integers $m$, and for all primes $p$ the number $x_{p^m}$ is divisible by $p$.
Consider a $m\times n$ rectangular board consisting of $mn$ unit squares. Two of its unit squares are called [i]adjacent[/i] if they have a common edge, and a [i]path[/i] is a sequence of unit squares in which any two consecutive squares are adjacent. Two parths are called [i]non-intersecting[/i] if they don't share any common squares.
Each unit square of the rectangular board can be colored black or white. We speak of a [i]coloring[/i] of the board if all its $mn$ unit squares are colored.
Let $N$ be the number of colorings of the board such that there exists at least one black path from the left edge of the board to its right edge. Let $M$ be the number of colorings of the board for which there exist at least two non-intersecting black paths from the left edge of the board to its right edge.
Prove that $N^{2}\geq M\cdot 2^{mn}$.
Do there exist polynomials $a(x)$, $b(x)$, $c(y)$, $d(y)$ such that \[1 + xy + x^2y^2= a(x)c(y) + b(x)d(y)\] holds identically?
Let $P=[a_{ij}]_{3\times 9}$ be a $3\times 9$ matrix where $a_{ij}\ge 0$ for all $i,j$. The following conditions are given:
[list][*]Every row consists of distinct numbers;
[*]$\sum_{i=1}^{3}x_{ij}=1$ for $1\le j\le 6$;
[*]$x_{17}=x_{28}=x_{39}=0$;
[*]$x_{ij}>1$ for all $1\le i\le 3$ and $7\le j\le 9$ such that $j-i\not= 6$.
[*]The first three columns of $P$ satisfy the following property $(R)$: for an arbitrary column $[x_{1k},x_{2k},x_{3k}]^T$, $1\le k\le 9$, there exists an $i\in\{1,2,3\}$ such that $x_{ik}\le u_i=\min (x_{i1},x_{i2},x_{i3})$.[/list]
Prove that:
a) the elements $u_1,u_2,u_3$ come from three different columns;
b) if a column $[x_{1l},x_{2l},x_{3l}]^T$ of $P$, where $l\ge 4$, satisfies the condition that after replacing the third column of $P$ by it, the first three columns of the newly obtained matrix $P'$ still have property $(R)$, then this column uniquely exists.
Let $A$ be a real $n\times n$ matrix satisfying
$$A+A^{\text T}=I,$$where $A^{\text T}$ denotes the transpose of $A$ and $I$ the $n\times n$ identity matrix. Show that $\det A>0$.
At the vertices $A, B, C, D, E, F, G, H$ of a cube, $2001, 2002, 2003, 2004, 2005, 2008, 2007$ and $2006$ stones respectively are placed. It is allowed to move a stone from a vertex to each of its three neighbours, or to move a stone to a vertex from each of its three neighbours. Which of the following arrangements of stones at $A, B, \ldots , H$ can be obtained?
$(\text{a})\quad 2001, 2002, 2003, 2004, 2006, 2007, 2008, 2005;$
$(\text{b})\quad 2002, 2003, 2004, 2001, 2006, 2005, 2008, 2007;$
$(\text{c})\quad 2004, 2002, 2003, 2001, 2005, 2008, 2007, 2006.$