Let $x_1, x_2, \dots, x_n$ be different real numbers. Prove that \[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll} 0, & \text { if } n \text { is even; } \\ 1, & \text { if } n \text { is odd. } \end{array}\right.\]

$A$ and $B$ are $2\times 2$ real valued matrices satisfying $$\det A = \det B = 1,\quad \text{tr}(A)>2,\quad \text{tr}(B)>2,\quad \text{tr}(ABA^{-1}B^{-1}) = 2$$ Prove that $A$ and $B$ have a common eigenvector.

Let $n \in \mathbb{N},n \ge 2,$ and $A,B \in \mathcal{M}_n(\mathbb{R}).$ Prove that there exists a complex number $z,$ such that $|z|=1$ and $$\Re \left( {\det(A+zB)} \right) \ge \det(A)+\det(B),$$ where $\Re(w)$ is the real part of the complex number $w.$

Given the determinant of order $n$ $$\begin{vmatrix} 8 & 3 & 3 & \dots & 3 \\ 3 & 8 & 3 & \dots & 3 \\ 3 & 3 & 8 & \dots & 3 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 3 & 3 & 3 & \dots & 8 \end{vmatrix}$$ Calculate its value and determine for which values of $n$ this value is a multiple of $10$.

Let $A, B \in M_2(\mathbb{R})$ such as $det(AB+BA)\leq 0$. Prove that $$det(A^2+B^2)\geq 0$$

Without calculating the value of the determinant $$ \begin{vmatrix}1 &1 &3& 1\\1& 2& 3 &5\\ 3& 0& 5& 5\\ 0& a& -11a& a^{13}+9a\end{vmatrix} , $$ show that it is divisible by $ 26, $ for any integer $ a. $

We consider a natural number $n$, $n \geq 2$ and the matrices \begin{tabular}{cc} $A= \begin{pmatrix} 1 & 2 & 3 & \cdots & n\\ n & 1 & 2 & \cdots & n - 1\\ n - 1 & n & 1 & \cdots & n - 2\\ \cdots & \cdots & \cdots & \cdots & \cdots\\2 & 3 & 4 & \cdots & 1 \end{pmatrix}$ \end{tabular} Show that$$\epsilon^ndet\left(I_n-A^{2n}\right)+\epsilon^{n-1}det\left(\epsilon I_n-A^{2n}\right)+\epsilon^{n-2}det\left(\epsilon^2 I_n-A^{2n}\right)+\cdots +det\left(\epsilon^n I_n-A^{2n}\right)$$ $$=n(-1)^{n-1}\left[\frac{n^n(n+1)}{2}\right]^{2n^2-4n}\left(1+(n+1)^{2n}\left(2n+(-1)^n{{2n}\choose{n}}\right)\right)$$where $\epsilon \in \mathbb{C}\backslash \mathbb{R}$, $\epsilon^{n+1}=1$

Let $n$ be a positive integer. Suppose that $A$ and $B$ are invertible $n \times n$ matrices with complex entries such that $A+B=I$ (where $I$ is the identity matrix) and \[(A^2+B^2)(A^4+B^4)=A^5+B^5.\] Find all possible values of $\det(AB)$ for the given $n$.

Let $A{}$ and $B$ be invertible $n\times n$ matrices with real entries. Suppose that the inverse of $A+B^{-1}$ is $A^{-1}+B$. Prove that $\det(AB)=1$. Does this property hold for $2\times 2$ matrices with complex entries?

Let $a,b, p_1 ,p_2, \ldots, p_n$ be real numbers with $a \ne b$. Define $f(x)= (p_1 -x) (p_2 -x) \cdots (p_n -x)$. Show that $$ \text{det} \begin{pmatrix} p_1 & a& a & \cdots & a \\ b & p_2 & a & \cdots & a\\ b & b & p_3 & \cdots & a\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ b & b& b &\cdots &p_n \end{pmatrix}= \frac{bf(a) -af(b)}{b-a}.$$

Prove that if $A{}$ and $B{}$ are $n\times n$ matrices with complex entries which satisfy \[A=AB-BA+A^2B-2ABA+BA^2+A^2BA-ABA^2,\]then $\det(A)=0$.

Let $n$ be a nonzero natural number and $f:\mathbb R\to\mathbb R\setminus\{0\}$ be a function such that $f(2014)=1-f(2013)$. Let $x_1,x_2,x_3,\ldots,x_n$ be real numbers not equal to each other. If $$\begin{vmatrix}1+f(x_1)&f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&1+f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&f(x_2)&1+f(x_3)&\cdots&f(x_n)\\\vdots&\vdots&\vdots&\ddots&\vdots\\f(x_1)&f(x_2)&f(x_3)&\cdots&1+f(x_n)\end{vmatrix}=0,$$prove that $f$ is not continuous.

Let $S_1, S_2, \dots, S_{2^n - 1}$ be the nonempty subsets of $\{1, 2, \dots, n\}$ in some order, and let $M$ be the $(2^n - 1) \times (2^n - 1)$ matrix whose $(i, j)$ entry is \[m_{ij} = \left\{ \begin{array}{cl} 0 & \text{if $S_i \cap S_j = \emptyset$}, \\ 1 & \text{otherwise}. \end{array} \right.\] Calculate the determinant of $M$.

Let $a$, $b$ and $c$ be complex numbers such that $abc = 1$. Find the value of the cubic root of \begin{tabular}{|ccc|} $b + n^3c$ & $n(c - b)$ & $n^2(b - c)$\\ $n^2(c - a)$ & $c + n^3a$ & $n(a - c)$\\ $n(b - a)$ & $n^2(a - b)$ & $a + n^3b$ \end{tabular}

Let be two matrices $ A,B\in M_2\left(\mathbb{R}\right) $ and two natural numbers $ m,n. $ Prove that: $$ \det\left( (AB)^m-(BA)^m\right)\cdot\det\left( (AB)^n-(BA)^n\right)\ge 0. $$

$n(\geq 2)$ is a given integer and $T$ is set of all $n\times n$ matrices whose entries are elements of the set $S=\{1,\cdots,2017\}$. Evaluate the following value. \[\sum_{A\in T} \text{det}(A)\]

Let $\mathcal M$ be the set of $n\times n$ matrices with integer entries. Find all $A\in\mathcal M$ such that $\det(A+B)+\det(B)$ is even for all $B\in\mathcal M$. [i]Proposed by Ethan Tan[/i]

In a new edition of QoTD duels, $n \ge 2$ ranked contestants (numbered 1 to $n$) play a round robin tournament (i.e. each pair of contestants compete against each other exactly once); no draws are possible. Define an upset to be a pair $(i, j)$ where$ i > j$ and contestant $i$ wins against contestant $j$. At the end of the tournament, contestant $i$ has $s_i$ wins for each $1 \le i \le n$. The result of the tournament is defined as the $n$-tuple $(s_1, s_2, \cdots , s_n)$. An $n$-tuple $S$ is called interesting if, among the distinct tournaments that produce $S$ as a result, the number of tournaments with an odd number of upsets is not equal to the number of tournaments with an even number of upsets. Find the number of interesting $n$-tuples in terms of $n$. [i](Two tournaments are considered distinct if the outcome of some match differs.)[/i]

Let $1=d_1<d_2<\ldots<d_{2m}=n$ be the divisors of a positive integer $n$, where $n$ is not a perfect square. Consider the determinant $$D=\begin{vmatrix}n+d_1&n&\ldots&n\\n&n+d_2&\ldots&n\\\ldots&\ldots&&\ldots\\n&n&\ldots&n+d_{2m}\end{vmatrix}.$$ (a) Prove that $n^m$ divides $D$. (b) Prove that $1+d_1+d_2+\ldots+d_{2m}$ divides $D$.

Let $n$ be a positive integer, and let $M_n$ be the set of invertible matrices with integer entries and size $n \times n$. (a) Find the largest possible value of $n$ such that there exists a symmetric matrix $A \in M_n$ satisfying \[ \det(A^{20} + A^{24}) < 2024. \] (b) Prove that for every $n$, there exists a matrix $B \in M_n$ such that \[ \det(B^{20} + B^{24}) < 2024. \]

Calculate the determinant of the $ n\times n $ complex matrix $ \left(a_j^i\right)_{1\le j\le n}^{1\le i\le n} $ defined by $$ a_j^i=\left\{\begin{matrix} 1+x^2,\quad i=j\\x,\quad |i-j|=1\\0,\quad |i-j|\ge 2\end{matrix}\right. , $$ where $ n $ is a natural number greater than $ 2. $

For any $n\times n$ unitary matrices $A,B$, prove that $|\det (A+2B)|\leq 3^n$.

Show that for any natural numbers $ m,n\ge 3, $ the equation $ \Delta_n (x)=0 $ has exactly two distinct solutions, where $$ \Delta_n (x)=\begin{vmatrix}1 & 1-m & 1-m & \cdots & 1-m & 1-m & -m \\ -1 & \binom{m}{x} & 0 & \cdots & 0 & 0 & 0 \\ 0 & -1 & \binom{m}{x} & \cdots & 0 & 0 & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & \cdots & -1 & \binom{m}{x} & 0 \\ 0 & 0 & 0 & \cdots & 0 & -1 & \binom{m}{x}\end{vmatrix} . $$

Let $x_1, x_2, \dots, x_n$ be different real numbers. Prove that \[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll} 0, & \text { if } n \text { is even; } \\ 1, & \text { if } n \text { is odd. } \end{array}\right.\]

Let $A=(\alpha_{ij})$ be a $m\,x\,n$ matric and $B=(\beta_{kl})$ be a $n\,x\, m$ matric with $m>n$ . Prove that $D(A\cdot B)=0$.