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]

Consider all $ m \times n$ matrices whose all entries are $ 0$ or $ 1$. Find the number of such matrices for which the number of $ 1$-s in each row and in each column is even.

Consider the set \[M = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in\mathcal{M}_2(\mathbb{C})\ |\ ab = cd \right\}.\] a) Give an example of matrix $A\in M$ such that $A^{2017}\in M$ and $A^{2019}\in M$, but $A^{2018}\notin M$. b) Show that if $A\in M$ and there exists the integer number $k\ge 1$ such that $A^k \in M$, $A^{k + 1}\in M$ si $A^{k + 2} \in M$, then $A^n\in M$, for any integer number $n\ge 1$.

Let $n$ be a positive integer that is not a perfect cube. Define real numbers $a,b,c$ by \[a=\root3\of n\kern1.5pt,\qquad b={1\over a-[a]}\kern1pt,\qquad c={1\over b-[b]}\kern1.5pt,\] where $[x]$ denotes the integer part of $x$. Prove that there are infinitely many such integers $n$ with the property that there exist integers $r,s,t$, not all zero, such that $ra+sb+tc=0$.

In a tournament, 2011 players meet 2011 times to play a multiplayer game. Every game is played by all 2011 players together and ends with each of the players either winning or losing. The standings are kept in two $2011\times 2011$ matrices, $T=(T_{hk})$ and $W=(W_{hk}).$ Initially, $T=W=0.$ After every game, for every $(h,k)$ (including for $h=k),$ if players $h$ and $k$ tied (that is, both won or both lost), the entry $T_{hk}$ is increased by $1,$ while if player $h$ won and player $k$ lost, the entry $W_{hk}$ is increased by $1$ and $W_{kh}$ is decreased by $1.$ Prove that at the end of the tournament, $\det(T+iW)$ is a non-negative integer divisible by $2^{2010}.$

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. $

Denote by $\mathcal M_2(\mathbb R)$ the set of all $2\times2$ matrices with real entries. Prove that: a) for every $A\in\mathcal M_2(\mathbb R)$ there exist $B,C\in\mathcal M_2(\mathbb R)$ such that $A=B^2+C^2$; b) there do not exist $B,C\in\mathcal M_2(\mathbb R)$ such that $\begin{pmatrix}0&1\\1&0\end{pmatrix}=B^2+C^2$ and $BC=CB$.

Let be a $ 3\times 3 $ complex matrix such that $ A^3=I $ and for which exist four real numbers $ a,b,c,d $ with $ a,c\neq 1 $ such that $ \det \left( A^2+aA+bI \right) =\det \left( A^2+cA+dI \right) =0. $ Show that $ a+b=c+d. $ [i]C. Merticaru[/i]

Let $k$ be a positive integer. Find the smallest positive integer $n$ for which there exists $k$ nonzero vectors $v_1,v_2,…,v_k$ in $\mathbb{R}^n$ such that for every pair $i,j$ of indices with $|i-j|>1$ the vectors $v_i$ and $v_j$ are orthogonal. [i]Proposed by Alexey Balitskiy, Moscow Institute of Physics and Technology and M.I.T.[/i]

Let $f_n\ (n=1,\ 2,\ \cdots)$ be a linear transformation expressed by a matrix $\left( \begin{array}{cc} 1-n & 1 \\ -n(n+1) & n+2 \end{array} \right)$ on the $xy$ plane. Answer the following questions: (1) Prove that there exists 2 lines passing through the origin $O(0,\ 0)$ such that all points of the lines are mapped to the same lines, then find the equation of the lines. (2) Find the area $S_n$ of the figure enclosed by the lines obtained in (1) and the curve $y=x^2$. (3) Find $\sum_{n=1}^{\infty} \frac{1}{S_n-\frac 16}.$ [i]2011 Tokyo Institute of Technlogy entrance exam, Problem 1[/i]

Let $\mathcal{F}$ be the set of pairs of matrices $(A,B)\in\mathcal{M}_2(\mathbb{Z})\times\mathcal{M}_2(\mathbb{Z})$ for which there exists some positive integer $k$ and matrices $C_1,C_2,\ldots, C_k\in\{A,B\}$ such that $C_1C_2\cdots C_k=O_2.$ For each $(A,B)\in\mathcal{F},$ let $k(A,B)$ denote the minimal positive integer $k$ which satisfies the latter property. [list=a] [*]Let $(A,B)\in\mathcal{F}$ with $\det(A)=0,\det(B)\neq 0$ and $k(A,B)=p+2$ for some $p\in\mathbb{N}^*.$ Show that $AB^pA=O_2.$ [*]Prove that for any $k\geq 3$ there exists a pair $(A,B)\in\mathcal{F}$ such that $k(A,B)=k.$ [/list][i]Bogdan Blaga[/i]

[color=darkred]A row of a matrix belonging to $\mathcal{M}_n(\mathbb{C})$ is said to be [i]permutable[/i] if no matter how we would permute the entries of that row, the value of the determinant doesn't change. Prove that if a matrix has two [i]permutable[/i] rows, then its determinant is equal to $0$ .[/color]

Solve in $ M_2(\mathbb{C})$ the equation $ X^2\equal{}\left( \begin{array}{cc} 1 & 2 \\ 3 & 6 \end{array} \right)$

In any cell of an $n \times n$ table a number is written such that all the rows are distinct. Prove that we can remove a column such that the rows in the new table are still distinct.

In rectangle $ ABCD$, points $ F$ and $ G$ lie on $ \overline{AB}$ so that $ AF \equal{} FG \equal{} GB$ and $ E$ is the midpoint of $ \overline{DC}$. Also, $ \overline{AC}$ intersects $ \overline{EF}$ at $ H$ and $ \overline{EG}$ at $ J$. The area of the rectangle $ ABCD$ is $ 70$. Find the area of triangle $ EHJ$. [asy] size(180); pair A, B, C, D, E, F, G, H, J; A = origin; real length = 6; real width = 3.5; B = length*dir(0); C = (length, width); D = width*dir(90); F = length/3*dir(0); G = 2*length/3*dir(0); E = (length/2, width); H = extension(A, C, E, F); J = extension(A, C, E, G); draw(A--B--C--D--cycle); draw(G--E--F); draw(A--C); label("$A$", A, dir(180)); label("$D$", D, dir(180)); label("$B$", B, dir(0)); label("$C$", C, dir(0)); label("$F$", F, dir(270)); label("$E$", E, dir(90)); label("$G$", G, dir(270)); label("$H$", H, dir(140)); label("$J$", J, dir(340)); [/asy] $ \displaystyle \textbf{(A)} \ \frac {5}{2} \qquad \textbf{(B)} \ \frac {35}{12} \qquad \textbf{(C)} \ 3 \qquad \textbf{(D)} \ \frac {7}{2} \qquad \textbf{(E)} \ \frac {35}{8}$

Let $ A$ be a unitary finite ring with $ n$ elements, such that the equation $ x^n\equal{}1$ has a unique solution in $ A$, $ x\equal{}1$. Prove that a) $ 0$ is the only nilpotent element of $ A$; b) there exists an integer $ k\geq 2$, such that the equation $ x^k\equal{}x$ has $ n$ solutions in $ A$.

Prove that for any two linear subspaces $V, W \subset R^n$ the same dimension there is an orthogonal transformation $A:R^n\to R^n$, such that $A(V )=W$ and $A(W) = V$

Let $p>2$ be a prime number and let $L=\{0,1,\dots,p-1\}^2$. Prove that we can find $p$ points in $L$ with no three of them collinear.

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 $n$ 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 $j \in \{1,2,3,\cdots,n\}$.

For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.

Let be a $ 2\times 2 $ real matrix $ A $ that has the property that $ \left| A^d-I_2 \right| =\left| A^d+I_2 \right| , $ for all $ d\in\{ 2014,2016 \} . $ Prove that $ \left| A^n-I_2 \right| =\left| A^n+I_2 \right| , $ for any natural number $ n. $

$x_{1},x_{2},\dots,x_{n}$ are real number such that for each $i$, the set $\{x_{1},x_{2},\dots,x_{n}\}\backslash \{x_{i}\}$ could be partitioned into two sets that sum of elements of first set is equal to the sum of the elements of the other. Prove that all of $x_{i}$'s are zero. [hide="Hint"]It is a number theory problem.[/hide]

Prove that these two statements are equivalent for an $n$ dimensional vector space $V$: [b]$\cdot$[/b] For the linear transformation $T:V\longrightarrow V$ there exists a base for $V$ such that the representation of $T$ in that base is an upper triangular matrix. [b]$\cdot$[/b] There exist subspaces $\{0\}\subsetneq V_1 \subsetneq ...\subsetneq V_{n-1}\subsetneq V$ such that for all $i$, $T(V_i)\subseteq V_i$.

$f: \mathbb R^{n}\longrightarrow\mathbb R^{m}$ is a non-zero linear map. Prove that there is a base $\{v_{1},\dots,v_{n}m\}$ for $\mathbb R^{n}$ that the set $\{f(v_{1}),\dots,f(v_{n})\}$ is linearly independent, after ommitting Repetitive elements.