Let $M$ be a set of real $n \times n$ matrices such that i) $I_{n} \in M$, where $I_n$ is the identity matrix. ii) If $A\in M$ and $B\in M$, then either $AB\in M$ or $-AB\in M$, but not both iii) If $A\in M$ and $B \in M$, then either $AB=BA$ or $AB=-BA$. iv) If $A\in M$ and $A \ne I_n$, there is at least one $B\in M$ such that $AB=-BA$. Prove that $M$ contains at most $n^2 $ matrices.

Find the Jordan normal form of the operator $e^{d/dt}$ in the space of quasi-polynomials $\{e^{\lambda t}p(t)\}$ where the degree of the polynomial $p$ is less than $5$, and of the operator $\text{ad}_A$, $B\mapsto [A, B]$, in the space of $n\times n$ matrices $B$, where $A$ is a diagonal matrix.

A real number with absolute value less than $1$ is written in each cell of an $n\times n$ array, so that the sum of the numbers in each $2\times 2$ square is zero. Show that for odd $n$ the sum of all the numbers is less than $n$.

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 $A_1, A_2,\dots,A_m\in \mathcal{M}_n(\mathbb{R})$. Prove that there exist $\varepsilon_1,\varepsilon_2,\dots,\varepsilon_m\in \{-1,1\}$ such that: $$\rm{tr}\left( (\varepsilon_1 A_1+\varepsilon_2A_2+\dots+\varepsilon_m A_m)^2\right)\geq \rm{tr}(A_1^2)+\rm{tr}(A_2^2)+\dots+\rm{tr}(A_m^2) $$

Prove that if a vector space is the union of some of it's proper subspaces, then number of these subspaces can not be less than the number of elements of the field of that vector space.

A square matrix is sum-magic if the sum of all elements in each row, column and major diagonal is constant. Similarly, a square matrix is product-magic if the product of all elements in each row, column and major diagonal is constant. Determine if there exist $3\times 3$ matrices of real numbers which are both sum-magic and product-magic.

Let $n$ be a positive integer divisible by $4$. Find the number of permutations $\sigma$ of $(1,2,3,\cdots,n)$ which satisfy the condition $\sigma(j)+\sigma^{-1}(j)=n+1$ for all $j \in \{1,2,3,\cdots,n\}$.

Define a function $w:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ as follows. For $|a|,|b|\le 2,$ let $w(a,b)$ be as in the table shown; otherwise, let $w(a,b)=0.$ \[\begin{array}{|lr|rrrrr|}\hline &&&&b&&\\ &w(a,b)&-2&-1&0&1&2\\ \hline &-2&-1&-2&2&-2&-1\\ &-1&-2&4&-4&4&-2\\ a&0&2&-4&12&-4&2\\ &1&-2&4&-4&4&-2\\ &2&-1&-2&2&-2&-1\\ \hline\end{array}\] For every finite subset $S$ of $\mathbb{Z}\times\mathbb{Z},$ define \[A(S)=\sum_{(\mathbf{s},\mathbf{s'})\in S\times S} w(\mathbf{s}-\mathbf{s'}).\] Prove that if $S$ is any finite nonempty subset of $\mathbb{Z}\times\mathbb{Z},$ then $A(S)>0.$ (For example, if $S=\{(0,1),(0,2),(2,0),(3,1)\},$ then the terms in $A(S)$ are $12,12,12,12,4,4,0,0,0,0,-1,-1,-2,-2,-4,-4.$)

Let $n$ be a positive integer, and $a,b$ be two complex numbers such that $a \neq 1$ and $b^k \neq 1$, for any $k \in \{1,2,\dots ,n\}$. The matrices $A,B \in \mathcal{M}_n(\mathbb{C})$ satisfy the relation $BA=a I_n + bAB$. Prove that $A$ and $B$ are invertible.

An $(n^2+n+1) \times (n^2+n+1)$ matrix of zeros and ones is given. If no four ones are vertices of a rectangle, prove that the number of ones does not exceed $(n + 1)(n^2 + n + 1).$

Let be two matrices $ A,B\in\mathcal{M}_2\left( \mathbb{R} \right) $ that don’t commute. [b]a)[/b] If $ A^3=B^3, $ then $ \text{tr} \left( A^n \right) =\text{tr} \left( B^n \right) , $ for all natural numbers $ n. $ [b]b)[/b] If $ A^n\neq B^n $ and $ \text{tr} \left( A^n \right) =\text{tr} \left( B^n \right) , $ for all natural numbers $ n, $ then find some of the matrices $ A,B. $

A triangle is bounded by the lines $a_1 x+ b_1 y +c_1=0$, $a_2 x+ b_2 y +c_2=0$ and $a_2 x+ b_2 y +c_2=0$. Show that its area, disregarding sign, is $$\frac{\Delta^{2}}{2(a_2 b_3- a_3 b_2)(a_3 b_1- a_1 b_3)(a_1 b_2- a_2 b_1)},$$ where $\Delta$ is the discriminant of the matrix $$M=\begin{pmatrix} a_1 & b_1 &c_1\\ a_2 & b_2 &c_2\\ a_3 & b_3 &c_3 \end{pmatrix}.$$

Let $\mathbb{M}$ be the vector space of $m \times p$ real matrices. For a vector subspace $S\subseteq \mathbb{M}$, denote by $\delta(S)$ the dimension of the vector space generated by all columns of all matrices in $S$. Say that a vector subspace $T\subseteq \mathbb{M}$ is a $\emph{covering matrix space}$ if \[ \bigcup_{A\in T, A\ne \mathbf{0}} \ker A =\mathbb{R}^p \] Such a $T$ is minimal if it doesn't contain a proper vector subspace $S\subset T$ such that $S$ is also a covering matrix space. [list] (a) (8 points) Let $T$ be a minimal covering matrix space and let $n=\dim (T)$ Prove that \[ \delta(T)\le \dbinom{n}{2} \] (b) (2 points) Prove that for every integer $n$ we can find $m$ and $p$, and a minimal covering matrix space $T$ as above such that $\dim T=n$ and $\delta(T)=\dbinom{n}{2}$[/list]

The cells of a $(n^2-n+1)\times(n^2-n+1)$ matrix are coloured using $n$ colours. A colour is called [i]dominant[/i] on a row (or a column) if there are at least $n$ cells of this colour on that row (or column). A cell is called [i]extremal[/i] if its colour is [i]dominant [/i] both on its row, and its column. Find all $n \ge 2$ for which there is a colouring with no [i]extremal [/i] cells. Iurie Boreico (Moldova)

Given 5 nonzero vectors in three-dimensional Euclidean space, prove that the sum of their pairwise angles is at most $6\pi$.

a)Let $A,B\in \mathcal{M}_3(\mathbb{R})$ such that $\text{rank}\ A>\text{rank}\ B$. Prove that $\text{rank}\ A^2\ge \text{rank}\ B^2$. b)Find the non-constant polynomials $f\in \mathbb{R}[X]$ such that $(\forall)A,B\in \mathcal{M}_4(\mathbb{R})$ with $\text{rank}\ A>\text{rank}\ B$, we have $\text{rank}\ f(A)>\text{rank}\ f(B)$.

Consider $A\in \mathcal{M}_{2020}(\mathbb{C})$ such that $$ (1)\begin{cases} A+A^{\times} =I_{2020},\\ A\cdot A^{\times} =I_{2020},\\ \end{cases} $$ where $A^{\times}$ is the adjugate matrix of $A$, i.e., the matrix whose elements are $a_{ij}=(-1)^{i+j}d_{ji}$, where $d_{ji}$ is the determinant obtained from $A$, eliminating the line $j$ and the column $i$. Find the maximum number of matrices verifying $(1)$ such that any two of them are not similar.

Each square of a $(2^n-1) \times (2^n-1)$ board contains either $1$ or $-1$. Such an arrangement is called [i]successful[/i] if each number is the product of its neighbors. Find the number of successful arrangements.

Let $A=(a_{ij})$ be the $n\times n$ matrix, where $a_{ij}$ is the remainder of the division of $i^j+j^i$ by $3$ for $i,j=1,2,\ldots,n$. Find the greatest $n$ for which $\det A\ne0$.

a magical $3\times3$ square is a $3\times3$ matrix containing all number from 1 to 9, and of which the sum of every row, every column, every diagonal, are all equal. Determine all magical $3\times3$ square

$f: \mathbb R^{n}\longrightarrow\mathbb R^{n}$ is a bijective map, that Image of every $n-1$-dimensional affine space is a $n-1$-dimensional affine space. 1) Prove that Image of every line is a line. 2) Prove that $f$ is an affine map. (i.e. $f=goh$ that $g$ is a translation and $h$ is a linear map.)

The difference between the cubes of two consecutive positive integers is equal to $n^2$ for a positive integer $n$. Show that $n$ is the sum of two squares.

Let $A=[a_{ij}]_{n \times n}$ be a $n \times n$ matrix whose elements are all numbers which belong to set $\{1,2,\cdots ,n\}$. Prove that by swapping the columns of $A$ with each other we can produce matrix $B=[b_{ij}]_{n \times n}$ such that $K(B) \le n$ where $K(B)$ is the number of elements of set $\{(i,j) ; b_{ij} =j\}$.

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 $(-1, 3, 2)$.