Let $\displaystyle{A}$ and $\displaystyle{B}$ be real symmetric matrixes with all eigenvalues strictly greater than $\displaystyle{1}$. Let $\displaystyle{\lambda }$ be a real eigenvalue of matrix $\displaystyle{{\rm A}{\rm B}}$. Prove that $\displaystyle{\left| \lambda \right| > 1}$. [i]Proposed by Pavel Kozhevnikov, MIPT, Moscow.[/i]

A $ 4\times 4$ table is divided into $ 16$ white unit square cells. Two cells are called neighbors if they share a common side. A [i]move[/i] consists in choosing a cell and the colors of neighbors from white to black or from black to white. After exactly $ n$ moves all the $ 16$ cells were black. Find all possible values of $ n$.

Determine all $2\times2$ integer matrices $A$ having the following properties: $1.$ the entries of $A$ are (positive) prime numbers, $2.$ there exists a $2\times2$ integer matrix $B$ such that $A=B^2$ and the determinant of $B$ is the square of a prime number.

Let be two natural numbers $ n\ge 2, k, $ and $ k\quad n\times n $ symmetric real matrices $ A_1,A_2,\ldots ,A_k. $ Then, the following relations are equivalent: $ 1)\quad \left| \sum_{i=1}^k A_i^2 \right| =0 $ $ 2)\quad \left| \sum_{i=1}^k A_iB_i \right| =0,\quad\forall B_1,B_2,\ldots ,B_k\in \mathcal{M}_n\left( \mathbb{R} \right) $ $ || $ [i]denotes the determinant.[/i]

Let be the sequence of sets $ \left(\left\{ A\in\mathcal{M}_2\left(\mathbb{R} \right) | A^{n+1} =2007^nA\right\}\right)_{n\ge 1} . $ [b]a)[/b] Prove that each term of the above sequence hasn't a finite cardinal. [b]b)[/b] Determine the intersection of the fourth element of the above sequence with the $ 2007\text{th} $ element. [i]Gheorghe Iurea[/i] [hide=Note]Similar with [url]https://artofproblemsolving.com/community/c7h1928039p13233629[/url].[/hide]

Let $n$ be a positive integer, $k\in \mathbb{C}$ and $A\in \mathcal{M}_n(\mathbb{C})$ such that $\text{Tr } A\neq 0$ and $$\text{rank } A +\text{rank } ((\text{Tr } A) \cdot I_n - kA) =n.$$ Find $\text{rank } A$.

For every integer $n\ge 3$ find all sequences of real numbers $(x_1,x_2,\ldots ,x_n)$ such that $\sum_{i=1}^{n}x_i=n$ and $\sum_{i=1}^{n} (x_{i-1}-x_i+x_{i+1})^2=n$, where $x_0=x_n$ and $x_{n+1}=x_1$.

Let $A=(\alpha_{ij})$ be a $m\,x\,n$ matric and $B=(\beta_{kl})$ be a $n\,x\, m$ matric with $m>n$ . Prove that $D(A\cdot B)=0$.

Let $A=(a_{ij})\in M_{(n+1)\times (n+1)}(\mathbb{R})$ with $a_{ij}=a+|i-j|d$, where $a$ and $d$ are fixed real numbers. Calculate $\det(A)$.

Let $A$ be a real $n \times n$ matrix and suppose that for every positive integer $m$ there exists a real symmetric matrix $B$ such that $$2021B = A^m+B^2.$$ Prove that $|\text{det} A| \leq 1$.

Let $M \in GL_{2n}(K)$, represented in block form as \[ M = \left[ \begin{array}{cc} A & B \\ C & D \end{array} \right] , M^{-1} = \left[ \begin{array}{cc} E & F \\ G & H \end{array} \right] \] Show that $\det M.\det H=\det A$.

$ABC$ is a triangle: $A=(0,0)$, $B=(36,15)$ and both the coordinates of $C$ are integers. What is the minimum area $\triangle ABC$ can have? $ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \frac{3}{2} \qquad\textbf{(D)}\ \frac{13}{2} \qquad\textbf{(E)}\ \text{there is no minimum} $

Let $I$ denote the $2\times2$ identity matrix $\begin{pmatrix}1&0\\0&1\end{pmatrix}$ and let $$M=\begin{pmatrix}I&A\\B&C\end{pmatrix},\enspace N=\begin{pmatrix}I&B\\A&C\end{pmatrix}$$where $A,B,C$ are arbitrary $2\times2$ matrices which entries in $\mathbb R$, the real numbers. Thus $M$ and $N$ are $4\times4$ matrices with entries in $\mathbb R$. Is it true that $M$ is invertible (i.e. there is a $4\times4$ matrix $X$ such that $MX=XM=I$) implies $N$ is invertible? Justify your answer.

$n$ teams take part in a tournament, in which every two teams compete exactly once, and that no draws are possible. It is known that for any two teams, there exists another team which defeated both of the two teams. Find all $n$ for which this is possible.

In any cell of an $n \times n$ table a number is written such that all the rows are distinct. Prove that we can remove a column such that the rows in the new table are still distinct.

Consider the set of all polynomials of degree less than or equal to $4$ with rational coefficients. a) Prove that it has a vector space structure over the field of numbers rational. b) Prove that the polynomials $1, x - 2, (x -2)^2, (x - 2)^3$ and $(x -2)^4$ form a base of this space. c) Express the polynomial $7 + 2x - 45x^2 + 3x^4$ in the previous base.

Let $P(x)$ and $Q(x)$ be arbitrary polynomials with real coefficients, and let $d$ be the degree of $P(x)$. Assume that $P(x)$ is not the zero polynomial. Prove that there exist polynomials $A(x)$ and $B(x)$ such that: (i) both $A$ and $B$ have degree at most $d/2$ (ii) at most one of $A$ and $B$ is the zero polynomial. (iii) $\frac{A(x)+Q(x)B(x)}{P(x)}$ is a polynomial with real coefficients. That is, there is some polynomial $C(x)$ with real coefficients such that $A(x)+Q(x)B(x)=P(x)C(x)$.

Given real numbers $x_i,y_i (i=1,2,\ldots ,n)$, let $A$ be the $n\times n$ matrix given by $a_{ij}=1$ if $x_i\ge y_j$ and $a_{ij}=0$ otherwise. Suppose $B$ is a $n\times n$ matrix whose entries are $0$ and $1$ such that the sum of entries in any row or column of $B$ equals the sum of entries in the corresponding row or column of $A$. Prove that $B=A$.

Let be the sequence of sets $ \left(\left\{ A\in\mathcal{M}_2\left(\mathbb{R} \right) | A^{n+1} =2003^nA\right\}\right)_{n\ge 1} . $ [b]a)[/b] Prove that each term of the above sequence hasn't a finite cardinal. [b]b)[/b] Determine the intersection of the third element of the above sequence with the $ 2003\text{rd} $ element. [i]Gheorghe Iurea[/i] [hide=Note]Similar with [url]https://artofproblemsolving.com/community/c7h1943241p13387495[/url].[/hide]

We know that $R^3 = \{(x_1, x_2, x_3) | x_i \in R, i = 1, 2, 3\}$ is a vector space regarding the laws of composition $(x_1, x_2, x_3) + (y_1, y_2, y_3) = (x_1 + y_1, x_2 + y_2, x_3 + y_3)$, $\lambda (x_1, x_2, x_3) = (\lambda x_1, \lambda x_2, \lambda x_3)$, $\lambda \in R$. We consider the following subset of $R^3$ : $L =\{(x_1, x2, x_3) \in R^3 | x_1 + x_2 + x_3 = 0\}$. a) Prove that $L$ is a vector subspace of $R^3$ . b) In $R^3$ the following relation is defined $\overline{x} R \overline{y} \Leftrightarrow \overline{x} -\overline{y} \in L, \overline{x} , \overline{y} \in R^3$. Prove that it is an equivalence relation. c) Find two vectors of $R^3$ that belong to the same class as the vector $(-1, 3, 2)$.

Let $v \in \mathbb{R}^2$ a vector of length 1 and $A$ a $2 \times 2$ matrix with real entries such that: (i) The vectors $A v, A^2 v$ y $A^3 v$ are also of length 1. (ii) The vector $A^2 v$ isn't equal to $\pm v$ nor to $\pm A v$. Prove that $A^t A=I_2$.

Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$. Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero. [i]Proposed by Kiran Kedlaya, USA[/i]

Two real square matrices $A$ and $B$ satisfy the conditions $A^{2002}=B^{2003}=I$ and $AB=BA$. Prove that $A+B+I$ is invertible. (The symbol $I$ denotes the identity matrix.)

Let $a_{i}$ and $b_{i}$ ($i=1,2, \cdots, n$) be rational numbers such that for any real number $x$ there is: \[x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}\] Find the least possible value of $n$.

Let be a natural number $ n\ge 2, $ the real numbers $ a_1,a_2,\ldots ,a_n,b_1,b_2,\ldots, b_n, $ and the matrix defined as $$ A=\left( a_i+b_j \right)_{1\le j\le n}^{1\le i\le n} . $$ [b]a)[/b] Show that $ n=2 $ if $ A $ is invertible. [b]b)[/b] Prove that the pair of numbers $ a_1,a_2 $ and $ b_1,b_2 $ are both consecutive (not necessarily in this order), if $ A $ is an invertible matrix of integers whose inverse is a matrix of integers. [i]Costică Ambrinoc[/i]