Let $K$ be a ring with unit and $M$ the set of $2 \times 2$ matrices constituted with elements of $K$. An addition and a multiplication are defined in $M$ in the usual way between arrays. It is requested to: a) Check that $M$ is a ring with unit and not commutative with respect to the laws of defined composition. b) Check that if $K$ is a commutative field, the elements of$ M$ that have inverse they are characterized by the condition $ad - bc \ne 0$. c) Prove that the subset of $M$ formed by the elements that have inverse is a multiplicative group.

Let $O(4,\mathbb Z)$ be the set of all $4\times4$ orthogonal matrices over $\mathbb Z$, i.e., $A^tA=I=AA^t$. Then $|O(4,\mathbb Z)|$ is

A matrix $\begin{bmatrix} x & y \\ z & w \end{bmatrix}$ has square root $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ if $$\begin{bmatrix} a & b \\ c & d \end{bmatrix}^2 = \begin{bmatrix} a^2 + bc &ab + bd \\ ac + cd & bc + d^2 \end{bmatrix} = \begin{bmatrix} x & y \\ z & w \end{bmatrix}$$ Determine how many square roots the matrix $\begin{bmatrix} 2 & 2 \\ 3 & 4 \end{bmatrix}$ has (complex coefficients are allowed).

$P\in A_n(\mathbb R)=\{M_{n\times n}|M^2=M\}$. Which of the following are true? $\textbf{(A)}~P^T=P,\forall P\in A_n(\mathbb R)$ $\textbf{(B)}~\exists P\ne0,P\in A_n(\mathbb R)\text{ with }\operatorname{tr}(P)=0$ $\textbf{(C)}~\exists X_{n\times r}\text{ such that }Px=X\text{ for }r=\operatorname{rank}(P)$

Given a square matrix $M$ of order $n$ over the field of numbers real, find, as a function of $M$, two matrices, one symmetric and one antisymmetric, such that their sum is precisely $ M$.

Define $S_0$ to be $1.$ For $n \geq 1 $, let $S_n $ be the number of $n\times n $ matrices whose elements are nonnegative integers with the property that $a_{ij}=a_{ji}$ (for $i,j=1,2,\ldots, n$) and where $\sum_{i=1}^{n} a_{ij}=1$ (for $j=1,2,\ldots, n$). Prove that a) $S_{n+1}=S_{n} +nS_{n-1}.$ b) $\sum_{n=0}^{\infty} S_{n} \frac{x^{n}}{n!} =\exp \left(x+\frac{x^{2}}{2}\right).$

Show that if $n\ge 2$ is an integer, then there exist invertible matrices $A_1, A_2, \ldots, A_n \in \mathcal{M}_2(\mathbb{R})$ with non-zero entries such that: \[A_1^{-1} + A_2^{-1} + \ldots + A_n^{-1} = (A_1 + A_2 + \ldots + A_n)^{-1}.\] [hide=Edit.] The $77777^{\text{th}}$ topic in College Math :coolspeak: [/hide]

Let $A, B \in \mathcal{M}_n(\mathbb{C})$ be such that $AB^2A = AB$. Prove that: a) $(AB)^2 = AB.$ b) $(AB - BA)^3 = O_n.$

Let $k$ and $n$ be positive integers. A sequence $\left( A_1, \dots , A_k \right)$ of $n\times n$ real matrices is [i]preferred[/i] by Ivan the Confessor if $A_i^2\neq 0$ for $1\le i\le k$, but $A_iA_j=0$ for $1\le i$, $j\le k$ with $i\neq j$. Show that $k\le n$ in all preferred sequences, and give an example of a preferred sequence with $k=n$ for each $n$. (Proposed by Fedor Petrov, St. Petersburg State University)

Let $ I $ and $ 0 $ be the square identity and null matrices, both of size $ 2019 $. There is a square matrix $A$ with rational entries and size $ 2019 $ such that: a) $ A ^ 3 + 6A ^ 2-2I = 0 $? b) $ A ^ 4 + 6A ^ 3-2I = 0 $?

Let \(A\) be a square matrix with entries in the field \(\mathbb Z / p \mathbb Z\) such that \(A^n - I\) is invertible for every positive integer \(n\). Prove that there exists a positive integer \(m\) such that \(A^m = 0\). [i](A matrix having entries in the field \(\mathbb Z / p \mathbb Z\) means that two matrices are considered the same if each pair of corresponding entries differ by a multiple of \(p\).)[/i] [i]Proposed by Tony Wang[/i]

Prove that for any positive integer $n$ there exists a matrix of the form $$A=\begin{pmatrix}1&a&b&c\\0&1&a&b\\0&0&1&a\\0&0&0&1\end{pmatrix},$$ (a) with nonzero entries, (b) with positive entries, such that the entries of $A^n$ are all perfect squares.

Find all natural numbers $ k\ge 1 $ and $ n\ge 2, $ which have the property that there exist two matrices $ A,B\in M_n\left(\mathbb{Z}\right) $ such that $ A^3=O_n $ and $ A^kB +BA=I_n. $

Let $ A\in M_2\left( \mathbb{C}\right) $ such that $ \det \left( A^2+A+I_2\right) =\det \left( A^2-A+I_2\right) =3. $ Prove that $ A^2\left( A^2+I_2\right) =2I_2. $

Here is a problem we (me and my colleagues) suggested and was given at the competition this year. The problem statement is very natural and short. However, we have not seen such a problem before. A real $2024 \times 2024$ matrix $A$ is called nice if $(Av, v) = 1$ for every vector $v\in \mathbb{R}^{2024}$ with unit norm. a) Prove that the only nice matrix such that all of its eigenvalues are real is the identity matrix. b) Find an example of a nice non-identity matrix

Let \( n \) be a positive integer, and let \( A \) and \( B \) be \( n \times n \) matrices with real coefficients such that \[ ABBA - BAAB = A - B. \] (a) Prove that \( \text{Tr}(A) = \text{Tr}(B) \) and that \( \text{Tr}(A^2) = \text{Tr}(B^2) \). (b) If \(BA^2B= A^2B^2\) and \(AB^2A= B^2A^2\), prove that \( \det A = \det B \). Note: \( \text{Tr}(X) \) denotes the trace of \( X \), which is the sum of the elements on its main diagonal, and \( \det X \) denotes the determinant of \( X \).

Let $S$ denote the set of $2$ by $2$ matrices with integer entries and determinant $1$, and let $T$ denote those matrices of $S$ which are congruent to the identity matrix $I\pmod3$ (so $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in T$ means that $a,b,c,d\in\mathbb Z,ad-bc=1,$ and $3$ divides $b,c,a-1,d-1$). (a) Let $f:T\to\mathbb R$ be a function such that for every $X,Y\in T$ with $Y\ne I$, either $f(XY)>f(X)$ or $f(XY^{-1})>f(X)$. Show that given two finite nonempty subsets $A,B$ of $T$, there are matrices $a\in A$ and $b\in B$ such that if $a'\in A$, $b'\in B$ and $a'b'=ab$, then $a'=a$ and $b'=b$. (b) Show that there is no $f:S\to\mathbb R$ such that for every $X,Y\in S$ with $Y\ne\pm I$, either $f(XY)>f(X)$ or $f(XY^{-1})>f(X)$.

Let $n$ be an integer with $n \geq 2$ and let $A \in \mathcal{M}_n(\mathbb{C})$ such that $\operatorname{rank} A \neq \operatorname{rank} A^2.$ Prove that there exists a nonzero matrix $B \in \mathcal{M}_n(\mathbb{C})$ such that $$AB=BA=B^2=0$$ [i]Cornel Delasava[/i]

Let $A, B \in M_2(\mathbb{R})$ such as $det(AB+BA)\leq 0$. Prove that $$det(A^2+B^2)\geq 0$$

For matrices $A=[a_{ij}]_{m \times m}$ and $B=[b_{ij}]_{m \times m}$ where $A,B \in \mathbb{Z} ^{m \times m}$ let $A \equiv B \pmod{n}$ only if $a_{ij} \equiv b_{ij} \pmod{n}$ for every $i,j \in \{ 1,2,...,m \}$, that's $A-B=nZ$ for some $Z \in \mathbb{Z}^{m \times m}$. (The symbol $A \in \mathbb{Z} ^{m \times m}$ means that every element in $A$ is an integer.) Prove that for $A \in \mathbb{Z} ^{m \times m}$ there is $B \in \mathbb{Z} ^{m \times m}$ , where $AB \equiv I \pmod{n }$ only if $(\det (A),n)=1$ and find the value of $B$ in the form of $A$ where $I$ represents the dimensional identity matrix $m \times m$. [i](PP-nine)[/i]

For any integer $n\ge 2$ and two $n\times n$ matrices with real entries $A,\; B$ that satisfy the equation $$A^{-1}+B^{-1}=(A+B)^{-1}\;$$ prove that $\det (A)=\det(B)$. Does the same conclusion follow for matrices with complex entries? (Proposed by Zbigniew Skoczylas, Wroclaw University of Technology)

Let $n\geqslant2$ be a natural number. Show that there is no real number $c{}$ for which \[\exp\left(\frac{T+S}{2}\right)\leqslant c\cdot \frac{\exp(T)+\exp(S)}{2}\]is satisfied for any self-adjoint $n\times n$ complex matrices $T{}$ and $S{}$. (If $A{}$ and $B{}$ are self-adjoint $n\times n$ matrices, $A\leqslant B$ means that $B-A$ is positive semi-definite.)

Let $ \mathcal{M} =\left\{ A\in M_2\left( \mathbb{C}\right)\big| \det\left( A-zI_2\right) =0\implies |z| < 1\right\} . $ Prove that: $$ X,Y\in\mathcal{M}\wedge X\cdot Y=Y\cdot X\implies X\cdot Y\in\mathcal{M} . $$