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]

Let $n\geq 2$ a fixed integer. We shall call a $n\times n$ matrix $A$ with rational elements a [i]radical[/i] matrix if there exist an infinity of positive integers $k$, such that the equation $X^k=A$ has solutions in the set of $n\times n$ matrices with rational elements. a) Prove that if $A$ is a radical matrix then $\det A \in \{-1,0,1\}$ and there exists an infinity of radical matrices with determinant 1; b) Prove that there exist an infinity of matrices that are not radical and have determinant 0, and also an infinity of matrices that are not radical and have determinant 1. [i]After an idea of Harazi[/i]

Compute the determinant of the $n\times n$ matrix $A=(a_{ij})_{ij}$, $$a_{ij}=\begin{cases} (-1)^{|i-j|} & \text{if}\, i\ne j,\\ 2 & \text{if}\, i= j. \end{cases}$$

Let $A$ be a $5\times10$ matrix with real entries, and let $A^{\text T}$ denote its transpose. Suppose every $5\times1$ matrix with real entries can be written in the form $A\mathbf u$ where $\mathbf u$ is a $10\times1$ matrix with real entries. Prove that every $5\times1$ matrix with real entries can be written in the form $AA^{\text T}\mathbf v$ where $\mathbf v$ is a $5\times1$ matrix with real entries.

Suppose a $3\times 3$ matrix $A$ satisfies $\mathbf{v}^t A \mathbf{v} > 0$ for any vector $\mathbf{v} \in\mathbb{R}^3 -\{0\}$. (Note that $A$ may not be a symmetric matrix.) (1) Prove that $\det(A)>0$. (2) Consider diagonal matrix $D=\text{diag}(-1,1,1)$. Prove that there's exactly one negative real among eigenvalues of $AD$.

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}$?

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 $a, b, c$ be integers. Prove that there are integers $p_1, q_1, r_1, p_2, q_2, r_2$ such that \[a = q_1r_2 - q_2r_1, b = r_1p_2 - r_2p_1, c = p_1q_2 - p_2q_1.\]

Let $S = \{1, \dots, n\}$. Given a bijection $f : S \to S$ an [i]orbit[/i] of $f$ is a set of the form $\{x, f(x), f(f(x)), \dots \}$ for some $x \in S$. We denote by $c(f)$ the number of distinct orbits of $f$. For example, if $n=3$ and $f(1)=2$, $f(2)=1$, $f(3)=3$, the two orbits are $\{1,2\}$ and $\{3\}$, hence $c(f)=2$. Given $k$ bijections $f_1$, $\ldots$, $f_k$ from $S$ to itself, prove that \[ c(f_1) + \dots + c(f_k) \le n(k-1) + c(f) \] where $f : S \to S$ is the composed function $f_1 \circ \dots \circ f_k$. [i]Proposed by Maria Monks Gillespie[/i]

Let $V$ be a 10-dimensional real vector space and $U_1,U_2$ two linear subspaces such that $U_1 \subseteq U_2, \dim U_1 =3, \dim U_2=6$. Let $\varepsilon$ be the set of all linear maps $T: V\rightarrow V$ which have $T(U_1)\subseteq U_1, T(U_2)\subseteq U_2$. Calculate the dimension of $\varepsilon$. (again, all as real vector spaces)

$n\geq 2$ is a given integer. For two permuations $(\alpha_1,\cdots,\alpha_n)$ and $(\beta_1,\cdots,\beta_n)$ of $1,\cdots,n$, consider $n\times n$ matrix $A= \left(a_{ij} \right)_{1\leq i,j\leq n}$ defined by $a_{ij} = (1+\alpha_i \beta_j )^{n-1}$. Find every possible value of $\det(A)$.

Let $ T$ be a bounded linear operator on a Hilbert space $ H$, and assume that $ \|T^n \| \leq 1$ for some natural number $ n$. Prove the existence of an invertible linear operator $ A$ on $ H$ such that $ \| ATA^{\minus{}1} \| \leq 1$. [i]E. Druszt[/i]

Let $m\ge 2$ be an integer, $A$ a finite set of integers (not necessarily positive) and $B_1,B_2,...,B_m$ subsets of $A$. Suppose that, for every $k=1,2,...,m$, the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $\dfrac{m}{2}$ elements.

Denote $\textrm{SL}_2 (\mathbb{Z})$ and $\textrm{SL}_3 (\mathbb{Z})$ the sets of matrices with $2$ rows and $2$ columns, respectively with $3$ rows and $3$ columns, with integer entries and their determinant equal to $1$. $\textbf{(a)}$ Let $N$ be a positive integer and let $g$ be a matrix with $3$ rows and $3$ columns, with rational entries. Suppose that for each positive divisor $M$ of $N$ there exists a rational number $q_M$, a positive divisor $f (M)$ of $N$ and a matrix $\gamma_M \in \textrm{SL}_3 (\mathbb{Z})$ such that \[ g = q_M \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & f (M) \end{array}\right) \gamma_M \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & M^{} \end{array}\right) . \] Moreover, if $q_1 = 1$, prove that $\det (g) = N$ and $g$ has the following shape: \[ g = \left(\begin{array}{ccc} a_{11} & a_{12} & Na_{13}\\ a_{21} & a_{22} & Na_{23}\\ Na_{31} & Na_{32} & Na_{33} \end{array}\right), \] where $a_{ij}$ are all integers, $i, j \in \{ 1, 2, 3 \} .$ $\textbf{(b)}$ Provide an example of a matrix $g$ with $2$ rows and $2$ columns which satisfies the following properties: $\bullet$ For each positive divisor $M$ of $6$ there exists a rational number $q_M$, a positive divisor $f (M)$ of $6$ and a matrix $\gamma_M \in \textrm{SL}_2 (\mathbb{Z})$ such that \[ g = q_M \left(\begin{array}{cc} 1 & 0\\ 0 & f (M) \end{array}\right) \gamma_M \left(\begin{array}{cc} 1 & 0\\ 0 & M^{} \end{array}\right) \] and $q_1 = 1$. $\bullet$ $g$ does not have its determinant equal to $6$ and is not of the shape \[ g = \left(\begin{array}{cc} a_{22} & 6 a_{23}\\ 6 a_{32} & 6 a_{33} \end{array}\right), \] where $a_{ij}$ are all positive integers, $i, j \in \{ 2, 3 \}$. [i](Radu Toma)[/i]

A sequence $a_n$ is defined by $\int_{a_n}^{a_{n+1}} (1+|\sin x|)dx=(n+1)^2\ (n=1,\ 2,\ \cdots),\ a_1=0$. Find $\lim_{n\to\infty} \frac{a_n}{n^3}$.

[list=a] [*]Let $A$ be a matrix from $\mathcal{M}_{2}(\mathbb{C})$, $A\neq aI_{2}$, for any $a\in\mathbb{C}$. Prove that the matrix $X$ from $\mathcal{M} _{2}(\mathbb{C})$ commutes with $A$, that is, $AX=XA$, if and only if there exist two complex numbers $\alpha$ and $\alpha^{\prime}$, such that $X=\alpha A+\alpha^{\prime}I_{2}$. [*]Let $A$, $B$ and $C$ be matrices from $\mathcal{M}_{2}(\mathbb{C})$, such that $AB\neq BA$, $AC=CA$ and $BC=CB$. Prove that $C$ commutes with all matrices from $\mathcal{M}_{2}(\mathbb{C})$.[/list]

Let $\mathcal{M}$ be the set of $5\times 5$ real matrices of rank $3$. Given a matrix in $\mathcal{M}$, the set of columns of $A$ has $2^5-1=31$ nonempty subsets. Let $k_A$ be the number of these subsets that are linearly independent. Determine the maximum and minimum values of $k_A$, as $A$ varies over $\mathcal{M}$. [i]The rank of a matrix is the dimension of the span of its columns.[/i]

Let $A$ be a $3\times 3$ real matrix such that the vectors $Au$ and $u$ are orthogonal for every column vector $u\in \mathbb{R}^{3}$. Prove that: a) $A^{T}=-A$. b) there exists a vector $v \in \mathbb{R}^{3}$ such that $Au=v\times u$ for every $u\in \mathbb{R}^{3}$, where $v \times u$ denotes the vector product in $\mathbb{R}^{3}$.

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]

How many solutions does the system have: $ \{\begin{matrix}&(3x+2y) *(\frac{3}{x}+\frac{1}{y})=2\\ & x^2+y^2\leq 2012\\ \end{matrix} $ where $ x,y $ are non-zero integers

An $n\times n$ matrix $A$ with integer entries is called [i]representative[/i] if, for any integer vector $\mathbf{v}$, there is a finite sequence $0=\mathbf{v}_0,\mathbf{v}_1,\dots,\mathbf{v}_{\ell}=\mathbf{v}$ of integer vectors such that for each $0\leq i <\ell$, either $\mathbf{v}_{i+1}=A\mathbf{v}_{i}$ or $\mathbf{v}_{i+1}-\mathbf{v}_i$ is an element of the standard basis (i.e. one of its entries is $1$, the rest are all equal to $0$). Show that $A$ is not representative if and only if $A^T$ has a real eigenvector with all non-negative entries and non-negative eigenvalue.

Let $n \in \mathbb N$, $n \geq 2$. (a) Give an example of two matrices $A,B \in \mathcal M_n \left( \mathbb C \right)$ such that \[ \textrm{rank} \left( AB \right) - \textrm{rank} \left( BA \right) = \left\lfloor \frac{n}{2} \right\rfloor . \] (b) Prove that for all matrices $X,Y \in \mathcal M_n \left( \mathbb C \right)$ we have \[ \textrm{rank} \left( XY \right) - \textrm{rank} \left( YX \right) \leq \left\lfloor \frac{n}{2} \right\rfloor . \] [i]Ion Savu[/i]

Let $n\geq 2$ be an integer and denote by $H_{n}$ the set of column vectors $^{T}(x_{1},\ x_{2},\ \ldots, x_{n})\in\mathbb{R}^{n}$, such that $\sum |x_{i}|=1$. Prove that there exist only a finite number of matrices $A\in\mathcal{M}_{n}(\mathbb{R})$ such that the linear map $f: \mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ given by $f(x)=Ax$ has the property $f(H_{n})=H_{n}$. [hide="Comment"]In the contest, the problem was given with a) and b): a) Prove the above for $n=2$; b) Prove the above for $n\geq 3$ as well.[/hide]

Ivan writes the matrix $\begin{pmatrix} 2 & 3\\ 2 & 4\end{pmatrix}$ on the board. Then he performs the following operation on the matrix several times: [b]1.[/b] he chooses a row or column of the matrix, and [b]2.[/b] he multiplies or divides the chosen row or column entry-wise by the other row or column, respectively. Can Ivan end up with the matrix $\begin{pmatrix} 2 & 4\\ 2 & 3\end{pmatrix}$ after finitely many steps?

Suppose a $n\times n$ real matrix $A$ satisfies $\text{tr}(A)=2018$, $\text{rank}(A)=1$. Prove that $A^2=2018 A$.