For an arbitrary square matrix $M$, define $$\exp(M)=I+\frac M{1!}+\frac{M^2}{2!}+\frac{M^3}{3!}+\ldots.$$Construct $2\times2$ matrices $A$ and $B$ such that $\exp(A+B)\ne\exp(A)\exp(B)$.

[b]Problem 1 [/b] Let $A$ and $B$ be two $3 \times 3$ matrices with real entries. Prove that $$ A-(A^{-1} +(B^{-1}-A)^{-1})^{-1} =ABA\ , $$ provided all the inverses appearing on the left-hand side of the equality exist.

Prove that the transformation product of the symmetry of center $(0, 0)$ with the symmetry of the axis, with the line of equation $x = y + 1$, can be expressed as a product of an axis symmetry the line $e$ by a translation of vector $\overrightarrow{v}$, with $e$ parallel to $\overrightarrow{v}$, . Determine a line $e$ and a vector $\overrightarrow{v}$, that meet the indicated conditions. have to be unique $e$ and $\overrightarrow{v}$,?

Let $ n$ be a positive integer, let the sets $ A_{1},A_{2},\cdots,A_{n \plus{} 1}$ be non-empty subsets of the set $ \{1,2,\cdots,n\}.$ prove that there exist two disjoint non-empty subsets of the set $ \{1,2,\cdots,n \plus{} 1\}$: $ \{i_{1},i_{2},\cdots,i_{k}\}$ and $ \{j_{1},j_{2},\cdots,j_{m}\}$ such that $ A_{i_{1}}\cup A_{i_{2}}\cup\cdots\cup A_{i_{k}} \equal{} A_{j_{1}}\cup A_{j_{2}}\cup\cdots\cup A_{j_{m}}$.

Let $n\in \mathbb{N}^*$ and a matrix $A\in \mathcal{M}_n(\mathbb{C}),\ A=(a_{ij})_{1\le i, j\le n}$ such that: \[a_{ij}+a_{jk}+a_{ki}=0,\ (\forall)i,j,k\in \{1,2,\ldots,n\}\] Prove that $\text{rank}\ A\le 2$.

Let $A$ be a real $n\times n$ Matrix and define $e^{A}=\sum_{k=0}^{\infty} \frac{A^{k}}{k!}$ Prove or disprove that for any real polynomial $P(x)$ and any real matrices $A,B$, $P(e^{AB})$ is nilpotent if and only if $P(e^{BA})$ is nilpotent.

a)Find a matrix $A\in \mathcal{M}_3(\mathbb{C})$ such that $A^2\neq O_3$ and $A^3=O_3$. b)Let $n,p\in\{2,3\}$. Prove that if there is bijective function $f:\mathcal{M}_n(\mathbb{C})\rightarrow \mathcal{M}_p(\mathbb{C})$ such that $f(XY)=f(X)f(Y),\ \forall X,Y\in \mathcal{M}_n(\mathbb{C})$, then $n=p$. [i]Ion Savu[/i]

Let $E = \{1, 2, \dots , 16\}$ and let $M$ be the collection of all $4 \times 4$ matrices whose entries are distinct members of $E$. If a matrix $A = (a_{ij} )_{4\times4}$ is chosen randomly from $M$, compute the probability $p(k)$ of $\max_i \min_j a_{ij} = k$ for $k \in E$. Furthermore, determine $l \in E$ such that $p(l) = \max \{p(k) | k \in E \}.$

Let $A$ be an $n\times n$ square matrix with integer entries. Suppose that $p^2A^{p^2}=q^2A^{q^2}+r^2I_n$ for some positive integers $p,q,r$ where $r$ is odd and $p^2=q^2+r^2$. Prove that $|\det A|=1$. (Here $I_n$ means the $n\times n$ identity matrix.)

Let $n, m \in \mathbb{N}$; $a_1,\ldots, a_m \in \mathbb{Z}^n$. Show that nonnegative integer linear combinations of these vectors give exactly the whole $\mathbb{Z}^n$ lattice, if $m \ge n$ and the following two statements are satisfied: [list] [*] The vectors do not fall into the half-space of $\mathbb{R}^n$ containing the origin (i.e. they do not fall on the same side of an $n-1$ dimensional subspace), [*] the largest common divisor (not pairwise, but together) of $n \times n$ minor determinants of the matrix $(a_1,\ldots, a_m)$ (which is of size $m \times n$ and the $i$-th column is $a_i$ as a column vector) is $1$. [/list]

There are some people in a meeting; each doesn't know at least 56 others, and for any pair, there exist a third one who knows both of them. Can the number of people be 65?

Let be two matrices $ A,B\in M_2\left(\mathbb{R}\right) $ that satisfy the equality $ \left( A-B\right)^2 =O_2. $ [b]a)[/b] Show that $ \det\left( A^2-B^2\right) =\left( \det A -\det B\right)^2. $ [b]b)[/b] Demonstrate that $ \det\left( AB-BA\right) =0\iff \det A=\det B. $

Let $ n$ and $ k$ be positive integers. Say that a permutation $ \sigma$ of $ \{1,2,\dots n\}$ is $ k$-[i]limited[/i] if $ |\sigma(i)\minus{}i|\le k$ for all $ i.$ Prove that the number of $ k$-limited permutations of $ \{1,2,\dots n\}$ is odd if and only if $ n\equiv 0$ or $ 1\pmod{2k\plus{}1}.$

Let $\{A_{1},\dots,A_{k}\}$ be matrices which make a group under matrix multiplication. Suppose $M=A_{1}+\dots+A_{k}$. Prove that each eigenvalue of $M$ is equal to $0$ or $k$.

Let be a natural number $ n\ge 2 $ and two $ n\times n $ complex matrices $ A,B $ that satisfy $ (AB)^3=O_n. $ Does this imply that $ (BA)^3=O_n ? $

Given a connected graph with $n$ edges, where there are no parallel edges. For any two cycles $C,C'$ in the graph, define its [i]outer cycle[/i] to be \[C*C'=\{x|x\in (C-C')\cup (C'-C)\}.\] (1) Let $r$ be the largest postive integer so that we can choose $r$ cycles $C_1,C_2,\ldots,C_r$ and for all $1\leq k\leq r$ and $1\leq i$, $j_1,j_2,\ldots,j_k\leq r$, we have \[C_i\neq C_{j_1}*C_{j_2}*\cdots*C_{j_k}.\] (Remark: There should have been an extra condition that either $j_1\neq i$ or $k\neq 1$) (2) Let $s$ be the largest positive integer so that we can choose $s$ edges that do not form a cycle. (Remark: A more precise way of saying this is that any nonempty subset of these $s$ edges does not form a cycle) Show that $r+s=n$. Note: A cycle is a set of edges of the form $\{A_iA_{i+1},1\leq i\leq n\}$ where $n\geq 3$, $A_1,A_2,\ldots,A_n$ are distinct vertices, and $A_{n+1}=A_1$.

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

Find all positive integers $x>1, y$ and primes $p,q$ such that $p^{x}=2^{y}+q^{x}$

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

Let $ n > 1$ be an odd positive integer and $ A = (a_{ij})_{i, j = 1..n}$ be the $ n \times n$ matrix with \[ a_{ij}= \begin{cases}2 & \text{if }i = j \\ 1 & \text{if }i-j \equiv \pm 2 \pmod n \\ 0 & \text{otherwise}\end{cases}.\] Find $ \det A$.

Let $G$ be the subgroup of $GL_{2}(\mathbb{R})$ generated by $A$ and $B$, where $$A=\begin{pmatrix} 2 &0\\ 0&1 \end{pmatrix},\; B=\begin{pmatrix} 1 &1\\ 0&1 \end{pmatrix}$$. Let $H$ consist of the matrices $\begin{pmatrix} a_{11} &a_{12}\\ a_{21}& a_{22} \end{pmatrix}$ in $G$ for which $a_{11}=a_{22}=1$. a) Show that $H$ is an abelian subgroup of $G$. b) Show that $H$ is not finitely generated.

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 $x_1,\ldots, x_n$ 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 $A=B$.

We are given the finite sets $ X$, $ A_1$, $ A_2$, $ \dots$, $ A_{n \minus{} 1}$ and the functions $ f_i: \ X\rightarrow A_i$. A vector $ (x_1,x_2,\dots,x_n)\in X^n$ is called [i]nice[/i], if $ f_i(x_i) \equal{} f_i(x_{i \plus{} 1})$, for each $ i \equal{} 1,2,\dots,n \minus{} 1$. Prove that the number of nice vectors is at least \[ \frac {|X|^n}{\prod\limits_{i \equal{} 1}^{n \minus{} 1} |A_i|}. \]

Let points $ A = (0,0) , \ B = (1,2), \ C = (3,3), $ and $ D = (4,0) $. Quadrilateral $ ABCD $ is cut into equal area pieces by a line passing through $ A $. This line intersects $ \overline{CD} $ at point $ \left (\frac{p}{q}, \frac{r}{s} \right ) $, where these fractions are in lowest terms. What is $ p + q + r + s $? $ \textbf{(A)} \ 54 \qquad \textbf{(B)} \ 58 \qquad \textbf{(C)} \ 62 \qquad \textbf{(D)} \ 70 \qquad \textbf{(E)} \ 75 $