# Questions tagged [determinant]

Questions about determinants: their computation or their theory. If $E$ is a vector space of dimension $d$, then we can compute the determinant of a $d$-uple $(v_1,\ldots,v_d)$ with respect to a basis.

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### binomial determinant

Let $n > 0$, then : $$\det \left( {2n \choose n+i-j} \right)_{i,j=0}^{n-1} = \prod_{i=0}^{n-1} \frac{2n+i \choose n}{n+i \choose n}$$ The LHS appears to be the determinant of a symetric ...
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### Is there a generalization of Pfaffians?

For an skew-symmetric matrix $A$ (meaning $A^T=-A$), the Pfaffian is defined by the equation $(\text{Pf}\,A)^2=\det A$. It is my understanding that this is defined for anti-symmetric matrices because ...
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### How to prove existence of a solution of this determinant equation?

Let $D\in\mathbb{R}^{n\times n}$ be a real diagonal matrix where $\sum_i D_{ii}<0$. Let also $R\in\mathbb{R}^{n\times n}$ and $L\in\mathbb{R}^{n\times n}$ be real (possibly) non-symmetric (...
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### Construct a matrix $M$ from $A$ and $B$ such that $\det(M)=\det(A)-\det(B)$

Given two $n \times n$ symmetric matrices $A$ and $B$, is there a generic way to construct a larger block matrix $M$ such that $\det(M) = \det(A) - \det(B)$? A simple block expression is desired, in ...
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### A conjecture concerning the irreducibility of characteristic polynomials of Arndt matrices

Letting $n \in \mathbb{N}$, let $M_{n}$ denote the $n \times n$ binary matrix with ones along the main antidiagonal and everywhere below the main antidiagonal and ones along the antidiagonal two ...
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### Determinant of a function

I was thinking about matrices and then why arent there matrices with uncountable many values? (Probably this conecpt already exists for a very long time, but i don't know it) Assume there are ...
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### Proving generalized Cassini's identity using determinant?

Motivation It is not hard to show, by using the general solution, that Proposition. If $(a_{n})_{n\in\Bbb{Z}}$ satisfies the recursive formula $a_{n+2} = pa_{n+1} + qa_{n}$, then for any $n, i, j$ ...
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### A challenge question in determinant of real matrices!

Suppose that $n\in \mathbb N -\{1\}$ and $a_{11},a_{12},\ldots,a_{nn}$ are $n^2$ distinct real numbers, prove that there is some enumeration of $a_{ij}$'s like $b_{ij}\ (i,j=1,2,\ldots,n)$ such that,...
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I have a $3n\times3n$ matrix $M$ that is in the following block tridiagonal form: $$M=\begin{pmatrix} A & B^T & 0\\ B & A & UBU \\ 0 & UB^T U & A\\ \end{pmatrix}$$ where \$A,B,...