Alice and Bob play the following game: Initially, there is a $2016\times2016$ "empty" matrix. Taking turns, with Alice playing first, each player chooses a real number and fill it into an empty entry. If the determinant of the last matrix is non-zero, then Alice wins. Otherwise, Bob wins. Who has the winning strategy?

Consider the sytem of equations \[ a_{11}x_1+a_{12}x_2+a_{13}x_3 = 0 \]\[a_{21}x_1+a_{22}x_2+a_{23}x_3 =0\]\[a_{31}x_1+a_{32}x_2+a_{33}x_3 = 0 \] with unknowns $x_1, x_2, x_3$. The coefficients satisfy the conditions: a) $a_{11}, a_{22}, a_{33}$ are positive numbers; b) the remaining coefficients are negative numbers; c) in each equation, the sum ofthe coefficients is positive. Prove that the given system has only the solution $x_1=x_2=x_3=0$.

Let $A,B,C\in \mathcal{M}_n(\mathbb{R})$ such that $ABC=O_n$ and $\text{rank}\ B=1$. Prove that $AB=O_n$ or $BC=O_n$.

Let be a natural number $ n\ge 2, $ a real number $ \lambda , $ and let be the set $$ H_{\lambda }=\left\{ \left( h_k^l \right)_{1\le k\le n}^{1\le l\le n}\in\mathcal{M}_n\left(\mathbb{R}\right) \bigg| \lambda =\sum_{k,l=1}^n h_k^l \right\} . $$ Prove the following statements. [b]a)[/b] The sets of symmetric and antisymmetric matrices from $ \mathcal{M}_n\left(\mathbb{R}\right) $ are subgroups of the additive subgroup $ \mathcal{M}_n\left(\mathbb{R}\right) , $ and any matrix from $ \mathcal{M}_n\left(\mathbb{R}\right) $ is a sum of a symmetric and antisymmetric matrix from $ \mathcal{M}_n\left(\mathbb{R}\right) . $ [b]b)[/b] $ \left( H_{\lambda },+\right)\le\left( \mathcal{M}_n\left(\mathbb{R}\right) ,+ \right)\iff \lambda =0 $ [b]c)[/b] There is a commutative group formed with the elements of $ H_{\lambda } $ if $ \lambda\neq 0. $ [i]Dan Negulescu[/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 $p$ be a prime. We arrange the numbers in ${\{1,2,\ldots ,p^2} \}$ as a $p \times p$ matrix $A = ( a_{ij} )$. Next we can select any row or column and add $1$ to every number in it, or subtract $1$ from every number in it. We call the arrangement [i]good[/i] if we can change every number of the matrix to $0$ in a finite number of such moves. How many good arrangements are there?

Let $Q$ be an $n$-by-$n$ real orthogonal matrix, and let $u\in \mathbb{R}^n$ be a unit column vector (that is, $u^Tu=1$). Let $P=I-2uu^T$, where $I$ is the $n$-by-$n$ identity matrix. Show that if $1$ is not an eigenvalue of $Q$, then $1$ is an eigenvalue of $PQ$.

Let $I$ denote the $2\times2$ identity matrix $\begin{pmatrix}1&0\\0&1\end{pmatrix}$ and let $$M=\begin{pmatrix}I&A\\B&C\end{pmatrix},\enspace N=\begin{pmatrix}I&B\\A&C\end{pmatrix}$$where $A,B,C$ are arbitrary $2\times2$ matrices which entries in $\mathbb R$, the real numbers. Thus $M$ and $N$ are $4\times4$ matrices with entries in $\mathbb R$. Is it true that $M$ is invertible (i.e. there is a $4\times4$ matrix $X$ such that $MX=XM=I$) implies $N$ is invertible? Justify your answer.

Hey, This problem is from the VTRMC 2006. 3. Recall that the Fibonacci numbers $ F(n)$ are defined by $ F(0) \equal{} 0$, $ F(1) \equal{} 1$ and $ F(n) \equal{} F(n \minus{} 1) \plus{} F(n \minus{} 2)$ for $ n \geq 2$. Determine the last digit of $ F(2006)$ (e.g. the last digit of 2006 is 6). As, I and a friend were working on this we noticed an interesting relationship when writing the Fibonacci numbers in "mod" notation. Consider the following, 01 = 1 mod 10 01 = 1 mod 10 02 = 2 mod 10 03 = 3 mod 10 05 = 5 mod 10 08 = 6 mod 10 13 = 3 mod 10 21 = 1 mod 10 34 = 4 mod 10 55 = 5 mod 10 89 = 9 mod 10 Now, consider that between the first appearance and second apperance of $ 5 mod 10$, there is a difference of five terms. Following from this we see that the third appearance of $ 5 mod 10$ occurs at a difference 10 terms from the second appearance. Following this pattern we can create the following relationships. $ F(55) \equal{} F(05) \plus{} 5({2}^{2})$ This is pretty much as far as we got, any ideas?

Calculate the sum of matrix commutators $[A, [B, C]] + [B, [C, A]] + [C, [A, B]]$, where $[A, B] = AB-BA$

Let $ A$ be an $ n \times n$ matrix whose elements are non-negative real numbers. Assume that $ A$ is a non-singular matrix and all elements of $ A^{\minus{}1}$ are non-negative real numbers. Prove that every row and every column of $ A$ has exactly one non-zero element.

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]

If $ n$ is a real number, then the simultaneous system $ nx \plus{} y \equal{} 1$ $ ny \plus{} z \equal{} 1$ $ x \plus{} nz \equal{} 1$ has no solution if and only if $ n$ is equal to $ \textbf{(A)}\ \minus{}1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 0 \text{ or } 1 \qquad \textbf{(E)}\ \frac12$

Given $ a_0 \equal{} 1$, $ a_1 \equal{} 3$, and the general relation $ a_n^2 \minus{} a_{n \minus{} 1}a_{n \plus{} 1} \equal{} (\minus{}1)^n$ for $ n \ge 1$. Then $ a_3$ equals: $ \textbf{(A)}\ \frac{13}{27}\qquad \textbf{(B)}\ 33\qquad \textbf{(C)}\ 21\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ \minus{}17$

Let $ABCD$ be a convex quadrilateral such that \[\begin{array}{rl} E,F \in [AB],& AE = EF = FB \\ G,H \in [BC],& BG = GH = HC \\ K,L \in [CD],& CK = KL = LD \\ M,N \in [DA],& DM = MN = NA \end{array}\] Let \[[NG] \cap [LE] = \{P\}, [NG]\cap [KF] = \{Q\},\] \[{[}MH] \cap [KF] = \{R\}, [MH]\cap [LE]=\{S\}\] Prove that [list=a][*]$Area(ABCD) = 9 \cdot Area(PQRS)$ [*] $NP=PQ=QG$ [/list]

Calculate $ \begin{pmatrix}1&0&0& \ldots &0\\\binom{1}{0} &\binom{1}{1} &0& \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ \binom{n}{0} &\binom{n}{1} & \binom{n}{2} & \ldots & \binom{n}{n}\end{pmatrix}^{-1} . $

There are $4$ distinct codes used in an intelligence station, one of them applied in each week. No two codes used in two adjacent weeks are the same code. Knowing that code $A$ is used in the first week, find the probability that code $A$ is used in the seventh week.

For a real coefficient cubic polynomial $f(x)=ax^3+bx^2+cx+d$, denote three roots of the equation $f(x)=0$ by $\alpha,\beta,\gamma$. Prove that the three roots $\alpha,\beta,\gamma$ are distinct real numbers iff the real symmetric matrix $$\begin{pmatrix} 3 & p_1 & p_2 \\ p_1 & p_2 & p_3 \\ p_2 & p_3 & p_4 \end{pmatrix},\quad p_i = \alpha^i + \beta^i + \gamma^i$$ is positive definite.

Justify your answer whether $A=\left( \begin{array}{ccc} -4 & -1& -1 \\ 1 & -2& 1 \\ 0 & 0& -3 \end{array} \right)$ is similar to $B=\left( \begin{array}{ccc} -2 & 1& 0 \\ -1 & -4& 1 \\ 0 & 0& -3 \end{array} \right),\ A,\ B\in{M(\mathbb{C})}$ or not.

Let $A,B \in \mathbb{R}^{n\times n}$ with $A^2+B^2=AB$. Prove that if $BA-AB$ is invertible then $3|n$.

Let be a polynom $ P $ of grade at least $ 2 $ and let be two $ 2\times 2 $ complex matrices such that $$ AB-BA\neq 0=P(AB)-P(BA). $$ Prove that there is a complex number $ \alpha $ having the property that $ P(AB)=\alpha I_2. $ [i]Titu Andreescu[/i] and [i]Dorin Andrica[/i]

Let $A,B\in \mathcal{M}_2(\mathbb{C})$ two non-zero matrices such that $AB+BA=O_2$ and $\det(A+B)=0$. Prove $A$ and $B$ have null traces.

We call an $n\times n$ matrix [i]well groomed[/i] if it only contains elements $0$ and $1$, and it does not contain the submatrix $\begin{pmatrix} 1& 0\\ 0 & 1 \end{pmatrix}.$ Show that there exists a constant $c>0$ such that every well groomed, $n\times n$ matrix contains a submatrix of size at least $cn\times cn$ such that all of the elements of the submatrix are equal. (A well groomed matrix may contain the submatrix $\begin{pmatrix} 0& 1\\ 1 & 0 \end{pmatrix}.$ )

For any square matrix $\mathcal{A}$ we define $\sin {\mathcal{A}}$ by the usual power series. \[ \sin {\mathcal{A}}=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}\mathcal{A}^{2n+1} \] Prove or disprove : $\exists 2\times 2$ matrix $A\in \mathcal{M}_2(\mathbb{R})$ such that \[ \sin{A}=\left(\begin{array}{cc}1 & 1996 \\0 & 1 \end{array}\right) \]

A [i]tanned vector[/i] is a nonzero vector in $\mathbb R^3$ with integer entries. Prove that any tanned vector of length at most $2021$ is perpendicular to a tanned vector of length at most $100$. [i]Proposed by Ethan Tan[/i]