3
votes
1answer
112 views

A variation of Cauchy's determinant

Prove the following identity: $$\det_{_{1\leq i,j \leq n}}\left(\frac{1}{(x_i+y_j)^2}\right)=\det_{_{1\leq i,j \leq n}}\left(\frac{1}{x_i+y_j}\right)\text{perm}_{_{1\leq i,j \leq ...
0
votes
2answers
42 views

Proving/disproving an identity on a Hessian.

Let $A=(a_{i,j})$ be a $n$ x $n$ matrix $(n\geq 2)$, where $a_{i,i} = |x|^2-2x_i ^2$ and $a_{i,j} = -2x_i x_j$ for $i\neq j$. Here $|x|^2 = x_1^2+x_2^2+ \cdots + x_n^2$. I'd like to compute the ...
2
votes
1answer
154 views

Number of zero entries in symmetric (0-1)-matrix with full diagonal

Let $S$ be an $n\times n$ symmetric matrix whose diagonal consists only of $1$s and whose other entries are either $0$ or $1$ . If the determinant and rank of $S$ are known, what can be said about ...
13
votes
2answers
205 views

Evaluate the determinant $\det\left[ \binom{2n}{n+i-j} \right]_{i,j=0}^{n-1}$

I am trying to show that: \begin{equation} \det\left[ \binom{2n}{n+i-j} \right]_{i,j=0}^{n-1}=\prod_{i=0}^{n-1} \frac{\binom{2n+i}{n}}{\binom{n+i}{n}} \end{equation} I have tried playing with the ...
2
votes
1answer
415 views

Proving determinant product rule combinatorially

One of definitions of the determinant is: $\det ({\mathbf C}) =\sum_{\lambda \in S_n} ({\operatorname {sgn} ({\lambda}) \prod_{k=1}^n C_{k \lambda ({k})}})$ I want to prove from this that ...
1
vote
1answer
140 views

Number of matrices with weakly increasing rows and columns

I'm curious as to how many matrices there are of size $m \times n$ with elements of the set $\{1, \ldots , k\}$ such that each row and column is weakly increasing? The answer should be expressable as ...
15
votes
4answers
591 views

Determinant of a generalized Pascal matrix

Let $M$ denote the infinite matrix defined recursively by $$ M_{ij} = \begin{cases} 1, & \text{if } i=1 \text{ and } j=1; \\ aM_{i-1,j}+bM_{i,j-1}+cM_{i-1,j-1}, & ...
10
votes
0answers
344 views

Prove this determinant identity combinatorially

This is for those of you who understand the Lindstrom-Gessel-Viennot lemma. I am looking for a proof of the following identity using paths and such: Let $A$ be an $n\times n$ matrix, and for ...
10
votes
1answer
348 views

Probability of a random $n \times n$ matrix over $\mathbb F_2$ being nonsingular

Given a random square matrix of size $n\times n$ in the field $\mathbb F_2$, what is the probability that its determinant is $1$? (This is also the probability that the matrix is non-singular, since ...
6
votes
1answer
419 views

Solving $n$-queens with determinants

I keep reading about a proposed method of finding solutions to the $n$-queens problem using determinants, but I can't find any specific details anywhere. Can somebody explain to me how to find ...