Let $A,B,C\in \mathcal{M}_n(\mathbb{R})$ such that $ABC=O_n$ and $\text{rank}\ B=1$. Prove that $AB=O_n$ or $BC=O_n$.

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 $ 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 $n \ge 1$ 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 $n \ge 2$ and $ a_1, a_2, \ldots , a_n $, nonzero real numbers not necessarily distinct. We define matrix $A = (a_{ij})_{1 \le i,j \le n} \in M_n( \mathbb{R} )$ , $a_{i,j} = max \{ a_i, a_j \}$, $\forall i,j \in \{ 1,2 , \ldots , n \} $. Show that $\mathbf{rank}(A) $= $\mathbf{card} $ $\{ a_k | k = 1,2, \ldots n \} $

Let $A\in M_3(\mathbb Z)$ such that $\det(A)=1$. What is the maximum possible number of entries of $A$ that are even?

Let $X_1,X_2,\ldots,X_m$ a numbering of the $m=2^n-1$ non-empty subsets of the set $\{1,2,\ldots,n\}$, $n\geq 2$. We consider the matrix $(a_{ij})_{1\leq i,j\leq m}$, where $a_{ij}=0$, if $X_i \cap X_j = \emptyset$, and $a_{ij}=1$ otherwise. Prove that the determinant $d$ of this matrix does not depend on the way the numbering was done and compute $d$.

Differentiable functions $f_1,\ldots,f_n:\mathbb R\to\mathbb R$ are linearly independent. Prove that there exist at least $n-1$ linearly independent functions among $f_1',\ldots,f_n'$.

The three row sums and the three column sums of the array \[\begin{bmatrix} 4 & 9 & 2 \\ 8 & 1 & 6 \\ 3 & 5 & 7 \end{bmatrix} \]are the same. What is the least number of entries that must be altered to make all six sums different from one another? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ 5$

Prove that there exists an infinite set of points \[ \dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots \] in the plane with the following property: For any three distinct integers $a,b,$ and $c$, points $P_a$, $P_b$, and $P_c$ are collinear if and only if $a+b+c=2014$.

If $ n$ is a real number, then the simultaneous system $ nx \plus{} y \equal{} 1$ $ ny \plus{} z \equal{} 1$ $ x \plus{} nz \equal{} 1$ has no solution if and only if $ n$ is equal to $ \textbf{(A)}\ \minus{}1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 0 \text{ or } 1 \qquad \textbf{(E)}\ \frac12$

If $A,B\in M_2(C)$ such that $AB-BA=B^2$ then prove that \[AB=BA\]

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 $A \in \mathcal{M}_2(\mathbb{C})$ and $C(A)=\{B \in \mathcal{M}_2(\mathbb{C}) : AB=BA \}.$ Prove that $$|\det(A+B)| \ge |\det B|,$$ for any $B \in C(A),$ if and only if $A^2=O_2.$

Nonnegative real numbers $p_{1},\ldots,p_{n}$ and $q_{1},\ldots,q_{n}$ are such that $p_{1}+\cdots+p_{n}=q_{1}+\cdots+q_{n}$ Among all the matrices with nonnegative entries having $p_i$ as sum of the $i$-th row's entries and $q_j$ as sum of the $j$-th column's entries, find the maximum sum of the entries on the main diagonal.

In the $xOy$ system, consider the collinear points $A_i(x_i,y_i),\ 1\le i\le 4$, such that there are invertible matrices $M\in \mathcal{M}_4(\mathbb{C})$ such that $(x_1,x_2,x_3,x_4)$ and $(y_1,y_2,y_3,y_4)$ are their first two lines. Prove that the sum of the entries of $M^{-1}$ doesn't depend of $M$. [i]Marian Andronache[/i]

$A = \begin{pmatrix} 2019 & 2020 & 2021 \\ 2020 & 2021 & 2022 \\ 2021 & 2022 & 2023 \end{pmatrix}$. Find $\text{rank}(A)$.

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 $A=B$.

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]

The following figure shows a [i]walk[/i] of length 6: [asy] unitsize(20); for (int x = -5; x <= 5; ++x) for (int y = 0; y <= 5; ++y) dot((x, y)); label("$O$", (0, 0), S); draw((0, 0) -- (1, 0) -- (1, 1) -- (0, 1) -- (-1, 1) -- (-1, 2) -- (-1, 3)); [/asy] This walk has three interesting properties: [list] [*] It starts at the origin, labelled $O$. [*] Each step is 1 unit north, east, or west. There are no south steps. [*] The walk never comes back to a point it has been to.[/list] Let's call a walk with these three properties a [i]northern walk[/i]. There are 3 northern walks of length 1 and 7 northern walks of length 2. How many northern walks of length 6 are there?

Let $n \ge 2$ be an integer, and let $O$ be the $n \times n$ matrix whose entries are all equal to $0$. Two distinct entries of the matrix are chosen uniformly at random, and those two entries are changed from $0$ to $1$. Call the resulting matrix $A$. Determine the probability that $A^2 = O$, as a function of $n$.

Let $ d_n$ be the determinant of the $ n\times n$ matrix whose entries, from left to right and then from top to bottom, are $ \cos 1,\cos 2,\dots,\cos n^2.$ (For example, $ d_3 \equal{} \begin{vmatrix}\cos 1 & \cos2 & \cos3 \\ \cos4 & \cos5 & \cos 6 \\ \cos7 & \cos8 & \cos 9\end{vmatrix}.$ The argument of $ \cos$ is always in radians, not degrees.) Evaluate $ \lim_{n\to\infty}d_n.$

Let $n$ be a positive integer, and let $M_n$ be the set of invertible matrices with integer entries and size $n \times n$. (a) Find the largest possible value of $n$ such that there exists a symmetric matrix $A \in M_n$ satisfying \[ \det(A^{20} + A^{24}) < 2024. \] (b) Prove that for every $n$, there exists a matrix $B \in M_n$ such that \[ \det(B^{20} + B^{24}) < 2024. \]

Let $A=\left( \begin{array}{ccc} 1 & 1& 0 \\ 0 & 1& 0 \\ 0 &0 & 2 \end{array} \right),\ B=\left( \begin{array}{ccc} a & 1& 0 \\ b & 2& c \\ 0 &0 & a+1 \end{array} \right)\ (a,\ b,\ c\in{\mathbb{C}}).$ (1) Find the condition for $a,\ b,\ c$ such that ${\text{rank} (AB-BA})\leq 1.$ (2) Under the condition of (1), find the condition for $a,\ b,\ c$ such that $B$ is diagonalizable.

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.