Found problems: 638
Let $v \in \mathbb{R}^2$ a vector of length 1 and $A$ a $2 \times 2$ matrix with real entries such that:
(i) The vectors $A v, A^2 v$ y $A^3 v$ are also of length 1.
(ii) The vector $A^2 v$ isn't equal to $\pm v$ nor to $\pm A v$.
Prove that $A^t A=I_2$.
Consider an $n \times n$ array of numbers $a_{ij}$ (standard notation). Suppose each row consists of the $n$ numbers $1,2,\ldots n$ in some order and $a_{ij} = a_{ji}$ for $i , j = 1,2, \ldots n$. If $n$ is odd, prove that the numbers $a_{11}, a_{22} , \ldots a_{nn}$ are $1,2,3, \ldots n$ in some order.
Let $A,B \in \mathcal{M}_n(\mathbb{C})$ be two matrices such that $A+B=AB+BA$. Prove that:
a) if $n$ is odd, then $\det(AB-BA)=0$;
b) if $\text{tr}(A)\neq \text{tr}(B)$, then $\det(AB-BA)=0$.
Let $f(z)=\frac{z+a}{z+b}$ and $g(z)=f(f(z))$, where $a$ and $b$ are complex numbers. Suppose that $|a|=1$ and $g(g(z))=z$ for all $z$ for which $g(g(z))$ is defined. What is the difference between the largest and smallest possible values of $|b|$?
$\textbf{(A)}\ 0 \qquad
\textbf{(B)}\ \sqrt{2}-1 \qquad
\textbf{(C)}\ \sqrt{3}-1 \qquad
\textbf{(D)}\ 1 \qquad
\textbf{(E)}\ 2$
Let $A\in\mathcal{M}_n(\mathbb{C})$ where $n\geq 2.$ Prove that if $m=|\{\text{rank}(A^k)-\text{rank}(A^{k+1})":k\in\mathbb{N}^*\}|$ then $n+1\geq m(m+1)/2.$
Determine all complex numbers $\lambda$ for which there exists a positive integer $n$ and a real $n\times n$ matrix $A$ such that $A^2=A^T$ and $\lambda$ is an eigenvalue of $A$.
Eight students solved $8$ problems.
a) It turned out that each problem was solved by $5$ students. Prove that there are two students such that each problem is solved by at least one of them.
b) If it turned out that each problem was solved by $4$ students, it can happen that there is no pair of students such that each problem is solved by at least one of them. (Give an example.)
Proposed by S. Tokarev
Let $A$ be an $n\times n$ matrix of real numbers for some $n\ge 1.$ For each positive integer $k,$ let $A^{[k]}$ be the matrix obtained by raising each entry to the $k$th power. Show that if $A^k=A^{[k]}$ for $k=1,2,\cdots,n+1,$ then $A^k=A^{[k]}$ for all $k\ge 1.$
Suppose that the automorphism group of the finite undirected graph $ X\equal{}(P, E)$ is isomorphic to the quaternion group (of order $ 8$). Prove that the adjacency matrix of $ X$ has an eigenvalue of multiplicity at least $ 4$.
($ P\equal{} \{ 1,2,\ldots, n \}$ is the set of vertices of the graph $ X$. The set of edges $ E$ is a subset of the set of all unordered pairs of elements of $ P$. The group of automorphisms of $ X$ consists of those permutations of $ P$ that map edges to edges. The adjacency matrix $ M\equal{}[m_{ij}]$ is the $ n \times n$ matrix defined by $ m_{ij}\equal{}1$ if $ \{ i,j \} \in E$ and $ m_{i,j}\equal{}0$ otherwise.)
[i]L. Babai[/i]
Given a $3 \times 3$ symmetric real matrix $A$, we define $f(A)$ as a $3 \times 3$ matrix with the same eigenvectors of $A$ such that if $A$ has eigenvalues $a$, $b$, $c$, then $f(A)$ has eigenvalues $b+c$, $c+a$, $a+b$ (in that order). We define a sequence of symmetric real $3\times3$ matrices $A_0, A_1, A_2, \ldots$ such that $A_{n+1} = f(A_n)$ for $n \geq 0$. If the matrix $A_0$ has no zero entries, determine the maximum number of indices $j \geq 0$ for which the matrix $A_j$ has any null entries.
Let be two matrices $ A,N\in\mathcal{M}_2(\mathbb{R}) $ that commute and such that $ N $ is nilpotent. Show that:
[b]a)[/b] $ \det (A+N)=\det (A) $
[b]b)[/b] if $ A $ is general linear, then the matrix $ A+N $ is invertible and $ (A+N)^{-1}=(A-N)A^{-2} . $
Show that there exists a $2 \times 2$ matrix of order 6 with rational entries, such that the sum of its entries is 2018.
Note: The order of a matrix (if it exists) is the smallest positive integer $n$ such that $A^n = I$, where $I$ is the identity matrix.
For any positive integer $ n$, let \[f(n) \equal{} \begin{cases} \log_8{n}, & \text{if }\log_8{n}\text{ is rational,} \\
0, & \text{otherwise.} \end{cases}\] What is $ \sum_{n \equal{} 1}^{1997}{f(n)}$?
$ \textbf{(A)}\ \log_8{2047}\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ \frac {55}{3}\qquad \textbf{(D)}\ \frac {58}{3}\qquad \textbf{(E)}\ 585$
For $ n\geq 0$ define the matrices $ A_n$ and $ B_n$ as follows: $ A_0 \equal{} B_0 \equal{} (1)$, and for every $ n>0$ let
\[ A_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\
A_{n \minus{} 1} & B_{n \minus{} 1} \\
\end{array} \right) \ \textrm{and} \ B_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\
A_{n \minus{} 1} & 0 \\
\end{array} \right).
\]
Denote by $ S(M)$ the sum of all the elements of a matrix $ M$. Prove that $ S(A_n^{k \minus{} 1}) \equal{} S(A_k^{n \minus{} 1})$, for all $ n,k\geq 2$.
We define a [i]pseudo-inverse[/i] $B\in \mathcal M_n(\mathbb C)$ of a matrix $A\in\mathcal M_n(\mathbb C)$ a matrix which fulfills the relations
\[ A = ABA \quad \text{ and } \quad B=BAB. \]
a) Prove that any square matrix has at least a pseudo-inverse.
b) For which matrix $A$ is the pseudo-inverse unique?
[i]Marius Cavachi[/i]
For an integer $n\geq 3$, let $\theta=2\pi/n$. Evaluate the determinant of the $n\times n$ matrix $I+A$, where $I$ is the $n\times n$ identity matrix and $A=(a_{jk})$ has entries $a_{jk}=\cos(j\theta+k\theta)$ for all $j,k$.
Let $ \mathbb{K} $ be a field and let be a matrix $ M\in\mathcal{M}_3(\mathbb{K} ) $ having the property that $ \text{tr} (A) =\text{tr} (A^2) =0 . $ Show that there is a $ \mu\in \mathbb{K} $ such that $ A^3=\mu A $ or $ A^3=\mu I. $
[i]Cristinel Mortici[/i]
Determine the number of possible values for the determinant of $A$, given that $A$ is a $n\times n$ matrix with real entries such that $A^3 - A^2 - 3A + 2I = 0$, where $I$ is the identity and $0$ is the all-zero matrix.
How many ordered pairs of positive integers $(x,y)$ are there such that $y^2-x^2=2y+7x+4$?
$ \textbf{(A)}\ 3
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 1
\qquad\textbf{(D)}\ 0
\qquad\textbf{(E)}\ \text{Infinitely many}
$
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 $O_n$ be the $n$-dimensional vector $(0,0,\cdots, 0)$. Let $M$ be a $2n \times n$ matrix of complex numbers such that whenever $(z_1, z_2, \dots, z_{2n})M = O_n$, with complex $z_i$, not all zero, then at least one of the $z_i$ is not real. Prove that for arbitrary real numbers $r_1, r_2, \dots, r_{2n}$, there are complex numbers $w_1, w_2, \dots, w_n$ such that
\[
\mathrm{re}\left[ M \left( \begin{array}{c} w_1 \\ \vdots \\ w_n \end{array}
\right) \right] = \left( \begin{array}{c} r_1 \\ \vdots \\ r_n
\end{array} \right).
\]
(Note: if $C$ is a matrix of complex numbers, $\mathrm{re}(C)$ is the matrix whose entries are the real parts of the entries of $C$.)
There are two matrices $A$ and $B$ of size $m\times n$ each filled only by “0”s and “1”s. It is given that along any row or column its elements do not decrease (from left to right and from top to bottom).It is also given that the numbers of “1”s in both matrices are equal and for any $k = 1, . . .$ , $m$ the sum of the elements in the top $k$ rows of the matrix $A$ is no less than that of the matrix $B$. Prove for any $l = 1, . . . $, $n$ the sum of the elements in left $l$ columns of the matrix $A$ is no greater than that of the matrix $B$.
Let $A$ and $B$ be complex Hermitian $2\times2$ matrices having the pairs of eigenvalues $(\alpha_1,\alpha_2)$ and $(\beta_1,\beta_2)$, respectively. Determine all possible pairs of eigenvalues $(\gamma_1,\gamma_2)$ of the matrix $C=A+B$. (We recall that a matrix $A=(a_{ij})$ is Hermitian if and only if $a_{ij}=\overline{a_{ji}}$ for all $i$ and $j$.)
Let be a natural number $ n\ge 2, $ the real numbers $ a_1,a_2,\ldots ,a_n,b_1,b_2,\ldots, b_n, $ and the matrix defined as
$$ A=\left( a_i+b_j \right)_{1\le j\le n}^{1\le i\le n} . $$
[b]a)[/b] Show that $ n=2 $ if $ A $ is invertible.
[b]b)[/b] Prove that the pair of numbers $ a_1,a_2 $ and $ b_1,b_2 $ are both consecutive (not necessarily in this order), if $ A $ is an invertible matrix of integers whose inverse is a matrix of integers.
[i]Costică Ambrinoc[/i]
Tessa the hyper-ant has a 2019-dimensional hypercube. For a real number $k$, she calls a placement of nonzero real numbers on the $2^{2019}$ vertices of the hypercube [i]$k$-harmonic[/i] if for any vertex, the sum of all 2019 numbers that are edge-adjacent to this vertex is equal to $k$ times the number on this vertex. Let $S$ be the set of all possible values of $k$ such that there exists a $k$-harmonic placement. Find $\sum_{k \in S} |k|$.