Let $A,B\in M_n(C)$ be two square matrices satisfying $A^2+B^2 = 2AB$. 1.Prove that $\det(AB-BA)=0$. 2.If $rank(A-B)=1$, then prove that $AB=BA$.

A $m\times n$ matrix $(a_{ij})$ of real numbers satisfies $|a_{ij}| <1$ and $\sum_{i=1}^m a_{ij}= 0$ for all$ j$. Show that one can permute the entries in each column in such a way that the obtained matrix $(b_{ij})$ satisfies $\sum_{j=1}^n b_{ij} < 2$ for all $i$.

Given positive integers $m,n$ and a $m\times n$ matrix $A$ with real entries. (1) Show that matrices $X = I_m + AA^T$ and $Y = I_n + A^T A$ are invertible. ($I_l$ is the $l\times l$ unit matrix.) (2) Evaluate the value of $\text{tr}(X^{-1}) - \text{tr}(Y^{-1})$.

Let $ A\equal{}(a_{ij})_{1\leq i,j\leq n}$ be a real $ n\times n$ matrix, such that $ a_{ij} \plus{} a_{ji} \equal{} 0$, for all $ i,j$. Prove that for all non-negative real numbers $ x,y$ we have \[ \det(A\plus{}xI_n)\cdot \det(A\plus{}yI_n) \geq \det (A\plus{}\sqrt{xy}I_n)^2.\]

Let $A=(a_{k,l})_{k,l=1,...,n}$ be a complex $n \times n$ matrix such that for each $m \in \{1,2,...,n\}$ and $1 \leq j_{1} <...<j_{m}$ the determinant of the matrix $(a_{j_{k},j_{l}})_{k,l=1,...,n}$ is zero. Prove that $A^{n}=0$ and that there exists a permutation $\sigma \in S_{n}$ such that the matrix $(a_{\sigma(k),\sigma(l)})_{k,l=1,...,n}$ has all of its nonzero elements above the diagonal.

Original in Hungarian; translated with Google translate; polished by myself. Let $n$ be a positive integer. Suppose that the sum of the matrices $A_1, \dots, A_n\in \Bbb R^{n\times n}$ is the identity matrix, but $\sum\nolimits_{i = 1}^n\alpha_i A_i$ is singular whenever at least one of the coefficients $\alpha_i \in \Bbb R$ is zero. a) Show that $\sum\nolimits_{i = 1}^n\alpha_i A_i$ is nonsingular if $\alpha_i\neq 0$ for all $i$. b) Show that if the matrices $A_i$ are symmetric, then all of them have rank $1$.

Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value. [i]Proposed by Marcin Kuczma, Poland[/i]

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 $k$ and $N$ be positive real numbers which satisfy $k\leq N$. For $1\leq i \leq k$, there are subsets $A_i$ of $\{1,2,3,\ldots,N\}$ that satisfy the following property. For arbitrary subset of $\{ i_1, i_2, \ldots , i_s \} \subset \{ 1, 2, 3, \ldots, k \} $, $A_{i_1} \triangle A_{i_2} \triangle ... \triangle A_{i_s}$ is not an empty set. Show that a subset $\{ j_1, j_2, .. ,j_t \} \subset \{ 1, 2, ... ,k \} $ exist that satisfies $n(A_{j_1} \triangle A_{j_2} \triangle \cdots \triangle A_{j_t}) \geq k$. ($A \triangle B=A \cup B-A \cap B$)

a) Let $A$ be a $n\times n$, $n\geq 2$, symmetric, invertible matrix with real positive elements. Show that $z_n\leq n^2-2n$, where $z_n$ is the number of zero elements in $A^{-1}$. b) How many zero elements are there in the inverse of the $n\times n$ matrix $$A=\begin{pmatrix} 1&1&1&1&\ldots&1\\ 1&2&2&2&\ldots&2\\ 1&2&1&1&\ldots&1\\ 1&2&1&2&\ldots&2\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 1&2&1&2&\ldots&\ddots \end{pmatrix}$$

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 $AD$.

$A,B \in M_{2\times 2}(C)$ Prove that: $Tr(AAABBABAABBB)=tr(BBBAABABBAAA)$

In the following $6\times 6$ matrix, one can choose any $k\times k$ submatrix, with $1<k\leq6 $ and add $1$ to all its entries. Is it possible to perform the operation a finite number of times so that all the entries in the $6\times 6$ matrix are multiples of $3$? $ \begin{pmatrix} 2 & 0 & 1 & 0 & 2 & 0 \\ 0 & 2 & 0 & 1 & 2 & 0 \\ 1 & 0 & 2 & 0 & 2 & 0 \\ 0 & 1 & 0 & 2 & 2 & 0 \\ 1 & 1 & 1 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} $ Note: A $p\times q$ submatrix of a $m\times n$ matrix (with $p\leq m$, $q\leq n$) is a $p\times q$ matrix formed by taking a block of the entries of this size from the original matrix.

Let $(G,\cdot)$ be a finite group with the identity element, $e$. The smallest positive integer $n$ with the property that $x^{n}= e$, for all $x \in G$, is called the [i]exponent[/i] of $G$. (a) For all primes $p \geq 3$, prove that the multiplicative group $\mathcal G_{p}$ of the matrices of the form $\begin{pmatrix}\hat 1 & \hat a & \hat b \\ \hat 0 & \hat 1 & \hat c \\ \hat 0 & \hat 0 & \hat 1 \end{pmatrix}$, with $\hat a, \hat b, \hat c \in \mathbb Z \slash p \mathbb Z$, is not commutative and has [i]exponent[/i] $p$. (b) Prove that if $\left( G, \circ \right)$ and $\left( H, \bullet \right)$ are finite groups of [i]exponents[/i] $m$ and $n$, respectively, then the group $\left( G \times H, \odot \right)$ with the operation given by $(g,h) \odot \left( g^\prime, h^\prime \right) = \left( g \circ g^\prime, h \bullet h^\prime \right)$, for all $\left( g,h \right), \, \left( g^\prime, h^\prime \right) \in G \times H$, has the [i]exponent[/i] equal to $\textrm{lcm}(m,n)$. (c) Prove that any $n \geq 3$ is the [i]exponent[/i] of a finite, non-commutative group. [i]Ion Savu[/i]

Let $A$ be a $n\times n$ complex matrix whose eigenvalues have absolute value at most $1$. Prove that $$ \|A^n\|\le \dfrac{n}{\ln 2} \|A\|^{n-1}. $$ (Here $\|B\|=\sup\limits_{\|x\|\leq 1} \|Bx\|$ for every $n\times n$ matrix $B$ and $\|x\|=\sqrt{\sum\limits_{i=1}^n |x_i|^2}$ for every complex vector $x\in\mathbb{C}^n$.) (Proposed by Ian Morris and Fedor Petrov, St. Petersburg State University)