Let $m$ and $n$ be integers greater than $1$ and $a_1 ,a_2 ,\ldots, a_{m+1}$ be real numbers. Prove that there exist real $n\times n$ matrices $A_1 ,A_2,\ldots, A_m$ such that (i) $\det(A_j) =a_j$ for $j=1,2,\ldots,m$ and (ii) $\det(A_1 +A_2 +\ldots+A_m)=a_{m+1}.$

Let $p \geq 3$ be a prime number and let $A$ be a matrix of order $p$ with complex entries. Assume that $\text{Tr}(A) = 0$ and $\det(A - I_p) \neq 0$. Prove that $A^p \neq I_p$. Note: $\text{Tr}(A)$ is the sum of the main diagonal elements of $A$ and $I_p$ is the identity matrix of order $p$.

We wish to construct a matrix with $19$ rows and $86$ columns, with entries $x_{ij} \in \{0, 1, 2\} \ (1 \leq i \leq 19, 1 \leq j \leq 86)$, such that: [i](i)[/i] in each column there are exactly $k$ terms equal to $0$; [i](ii)[/i] for any distinct $j, k \in \{1, . . . , 86\}$ there is $i \in \{1, . . . , 19\}$ with $x_{ij} + x_{ik} = 3.$ For what values of $k$ is this possible?

a) Let $A$ be a $n\times n$, $n\geq 2$, symmetric, invertible matrix with real positive elements. Show that $z_n\leq n^2-2n$, where $z_n$ is the number of zero elements in $A^{-1}$. b) How many zero elements are there in the inverse of the $n\times n$ matrix $$A=\begin{pmatrix} 1&1&1&1&\ldots&1\\ 1&2&2&2&\ldots&2\\ 1&2&1&1&\ldots&1\\ 1&2&1&2&\ldots&2\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 1&2&1&2&\ldots&\ddots \end{pmatrix}$$

Let $n \ge 1$ be a positive integer. An $n \times n$ matrix is called [i]good[/i] if each entry is a non-negative integer, the sum of entries in each row and each column is equal. A [i]permutation[/i] matrix is an $n \times n$ matrix consisting of $n$ ones and $n(n-1)$ zeroes such that each row and each column has exactly one non-zero entry. Prove that any [i]good[/i] matrix is a sum of finitely many [i]permutation[/i] matrices.

Let be a real $ 4\times 4 $ real matrix with $ \text{det} \left( A^2-I\right) <0. $ Prove that there is a number $ \alpha\in (-1,1) $ so that $ A+\alpha I $ is singular. [i]Mihai Haivas[/i]

The matrix \[A=\begin{pmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ldots & \vdots \\ a_{n1} & \ldots & a_{nn} \end{pmatrix}\] satisfies the inequality $\sum_{j=1}^n |a_{j1}x_1 + \cdots+ a_{jn}x_n| \leq M$ for each choice of numbers $x_i$ equal to $\pm 1$. Show that \[|a_{11} + a_{22} + \cdots+ a_{nn}| \leq M.\]

Consider an infinite sequence $a_1,a_2,\cdots$ whose terms all belong to $\left\{1,2\right\}$. A positive integer with $n$ digits is said to be [i]good[/i] if its decimal representation has the form $a_ra_{r+1}\cdots a_{r+(n-1)}$, for some positive integer $r$. Suppose that there are at least $2008$ [i]good[/i] numbers with a million digits. Prove that there are at least $2008$ [i]good[/i] numbers with $2007$ digits.

Let \( A \) be a \( 2 \times 2 \) matrix with integer entries and \(\det A \neq 0\). If the sequence \(\operatorname{tr}(A^n)\), for \( n = 1, 2, 3, \ldots \), is bounded, show that \[ A^{12} = I \quad \text{or} \quad (A^2 - I)^2 = O. \] Here, \( I \) and \( O \) denote the identity and zero matrices, respectively, and \(\operatorname{tr}\) denotes the trace of the matrix (the sum of the elements on the main diagonal).

Let $A,B\in\mathcal{M}_{2}(\mathbb{R})$ (real $2\times 2$ matrices), that satisfy $A^{2}+B^{2}=AB$. Prove that $(AB-BA)^{2}=O_{2}$.

Consider the two square matrices \[A=\begin{bmatrix} +1 & +1 &+1& +1 &+1 \\+1 &+1 &+1&-1 &-1 \\ +1 &-1&-1 &+1& +1 \\ +1 & -1 & -1 & -1 & +1 \\ +1 &+1&-1 &+1&-1 \end{bmatrix} \quad \text{ and } \quad B=\begin{bmatrix} +1 & +1 &+1& +1 &+1 \\+1 &+1 &+1&-1 &-1 \\ +1 &+1&-1& +1&-1 \\ +1 &-1& -1& +1& +1 \\ +1 & -1& +1&-1 &+1 \end{bmatrix}\] with entries $+1$ and $-1$. The following operations will be called elementary: (1) Changing signs of all numbers in one row; (2) Changing signs of all numbers in one column; (3) Interchanging two rows (two rows exchange their positions); (4) Interchanging two columns. Prove that the matrix $B$ cannot be obtained from the matrix $A$ using these operations.

Let $ ABC$ be an isosceles right triangle and $M$ be the midpoint of its hypotenuse $AB$. Points $D$ and $E$ are taken on the legs $AC$ and $BC$ respectively such that $AD=2DC$ and $BE=2EC$. Lines $AE$ and $DM$ intersect at $F$. Show that $FC$ bisects the $\angle DFE$.

2500 chess kings have to be placed on a $100 \times 100$ chessboard so that [b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex); [b](ii)[/b] each row and each column contains exactly 25 kings. Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.) [i]Proposed by Sergei Berlov, Russia[/i]

Assume $m\times n$ matrix which is filled with just 0, 1 and any two row differ in at least $n/2$ members, show that $m \leq 2n$. ( for example the diffrence of this two row is only in one index 110 100) [i]Edited by Myth[/i]

The points $(0,0),$ $(a,11)$, and $(b,37)$ are the vertices of an equilateral triangle. Find the value of $ab$.

There are two matrices $A$ and $B$ of size $m\times n$ each filled only by “0”s and “1”s. It is given that along any row or column its elements do not decrease (from left to right and from top to bottom).It is also given that the numbers of “1”s in both matrices are equal and for any $k = 1, . . .$ , $m$ the sum of the elements in the top $k$ rows of the matrix $A$ is no less than that of the matrix $B$. Prove for any $l = 1, . . . $, $n$ the sum of the elements in left $l$ columns of the matrix $A$ is no greater than that of the matrix $B$.

Let $A=(a_{ij})_{i, j=1}^n$ be a symmetric $n\times n$ matrix with real entries, and let $\lambda _1, \lambda _2, \dots, \lambda _n$ denote its eigenvalues. Show that $$\sum_{1\le i<j\le n} a_{ii}a_{jj}\ge \sum_{1\le i < j\le n} \lambda _i \lambda _j$$ and determine all matrices for which equality holds. (Proposed by Matrin Niepel, Comenius University, Bratislava)

Sequences $(x_n)_{n\ge1}$, $(y_n)_{n\ge1}$ satisfy the relations $x_n=4x_{n-1}+3y_{n-1}$ and $y_n=2x_{n-1}+3y_{n-1}$ for $n\ge1$. If $x_1=y_1=5$ find $x_n$ and $y_n$. Calculate $\lim_{n\rightarrow\infty}\frac{x_n}{y_n}$.

Let $A$ be an invertible $n \times n$ matrix with complex entries. Suppose that for each positive integer $m$, there exists a positive integer $k_m$ and an $n \times n$ invertible matrix $B_m$ such that $A^{k_m m} = B_m A B_m ^{-1}$. Show that all eigenvalues of $A$ are equal to $1$.

Let $d\leq n$ be positive integers and $A$ a real $d\times n$ matrix. Let $\sigma(A)$ be the supremum of $\inf_{v\in W,|v|=1}|Av|$ over all subspaces $W$ of $R^n$ with dimension $d$. For each $j \leq d$, let $r(j) \in \mathbb{R}^n$ be the $j$th row vector of $A$. Show that: \[\sigma(A) \leq \min_{i\leq d} d(r(i), \langle r(j), j\ne i\rangle) \leq \sqrt{n}\sigma(A)\] In which all are euclidian norms and $d(r(i), \langle r(j), j\ne i\rangle)$ denotes the distance between $r(i)$ and the span of $r(j), 1 \leq j \leq d, j\ne i$.

Let $A,B\in M_n(C)$ be two square matrices satisfying $A^2+B^2 = 2AB$. 1.Prove that $\det(AB-BA)=0$. 2.If $rank(A-B)=1$, then prove that $AB=BA$.

Let be a natural number $ n\ge 2 $ and the $ n\times n $ matrix whose entries at the $ \text{i-th} $ line and $ \text{j-th} $ column is $ \min (i,j) . $ Calculate: [b]a)[/b] its determinant. [b]b)[/b] its inverse.

Show that if $k\leq \frac n2$ and $\mathcal F$ is a family $k\times k$ submatrices of an $n\times n$ matrix such that any two intersect then $$|\mathcal F|\leq \binom{n-1}{k-1}^2$$ [Gy. Katona]

A matrix $M$ is called idempotent if $M^2 = M$. Find an idempotent $2 \times 2$ matrix with distinct rational entries or write “none” if none exist.

Let $S$ be a set of $ 2 \times 2 $ integer matrices whose entries $a_{ij}(1)$ are all squares of integers and, $(2)$ satisfy $a_{ij} \le 200$. Show that $S$ has more than $ 50387 (=15^4-15^2-15+2) $ elements, then it has two elements that commute.