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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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Determinant of the matrix $I+A+A^2+ \cdots + A^{n-1}$.

Let $c_1, \ldots, c_n$ be real numbers and let $A = (a_{ij})_{n\times n}$ the matrix defined by $a_{ii+1} = c_i$ for each $i = 1,2, \ldots, n-1$, $a_{n1} = c_{n}$ and the other entries are zero. Show ...
Mauricio Urrego Vasquez's user avatar
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Inverse and Determinant of Matrix $Axx^TA+cA$

Fix $c \in \mathbb{R}$, a symmetric (if needed, positive definite) $n \times n$ real matrix $A$, and $x \in \mathbb{R}^{n \times 1}$. I need help computing the determinant and inverse of the $n \times ...
温泽海's user avatar
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Query regarding determinants - getting different results from different methods.

$ \begin{vmatrix} (1+a) & 1 & 1 & 1 \\ 1 &(1+b)& 1 & 1 \\ 1 & 1 &(1+c) & 1 \\ 1 &1 & 1 &(1+d) \\ \end{vmatrix} $ This determinant is solved ...
Qwerty's user avatar
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Prove $D_3 = (\det{\Delta})^2$

Given 3 vectors in ${\mathbb{R}^3}$: $${{\vec \alpha }_1} = \left( {{a_1},{a_2},{a_3}} \right),{{\vec \alpha }_2} = \left( {{b_1},{b_2},{b_3}} \right),{{\vec \alpha }_3} = \left( {{c_1},{c_2},{c_3}} \...
dienhosp3's user avatar
  • 573
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Determinant of a similarity matrix

I have a set of vectors, and a pairwise similarity matrix defined like: $$ K_{ij} = e^{-\frac{1}{2}\|x_i - x_j\|^2} $$ Where each $x$ is a vector. I want to know if there is any interpretation of the ...
Mercury's user avatar
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Range of values of θ that satisy these values.

The question:The value of θ for which $$x+y(\sin θ)=1, x(\sin θ)+4y=2$$ satisfy $$x>=\frac{4}{5}, y>=\frac{1}{3}$$ Note here θ must belong from $$(-\frac{\pi}{2},\frac{\pi}{2})$$ This is a ...
Mahit Chopra's user avatar
-4 votes
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Does $\det(kA-B)= k^n\det(A)-\det(B) hold? [closed]

I have a doubt in matrix..does $\det(kA-B)= k^n\det(A)-det(B)$ always hold?.or there are some special cases?.If $A/B$ are symmetric or skew symmetric then we can do something like this?
Choco's user avatar
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13 votes
2 answers
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Does anyone know a non-trivial surjective multiplicative homomorphism from the $4\times 4$ matrices to the $2 \times 2 $ matrices?

It's well known the determinant provides a surjective homomorphism between $n \times n$ real matrices and $\mathbb{R}$ with the operation of multiplication. I was curious of inter-matrix ...
Sidharth Ghoshal's user avatar
-3 votes
2 answers
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Prove that the determinant of a matrix with a+b on diagonals and a on off diagonals is (b^(n-1))(na+b) [closed]

Prove that a square nxn matrix's determinant is equal to (b^(n-1))(na+b)
Saevpatoria's user avatar
-1 votes
1 answer
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Question in Linear Algebra regarding a Gram matrix

I will first provide the question: Let V be an n dimensional inner product space. let v1,...,vn ∈ V and let (A)ij = (<vi,vj>). prove: det(A) = 0 if and only if v1,...,vn are linearly dependent. ...
no name's user avatar
-2 votes
0 answers
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Rewriting determinant in proof of existence of permutation of bases.

I am currently reviewing linear algebra from Steven Roman's Advanced Linear Algebra, and I have gotten stuck on problem 1.23 for quite some time. The question asks given bases $B = \{b_1, ..., b_n\}, ...
tigs's user avatar
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1 answer
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Which linear transformations on hermitian 2x2 matrices fix the determinant?

We consider the four dimensional real vector space $H$ of all $2\times 2$ hermitian matrices. The determinant is a function $\det\colon H\to\mathbb{R}$. I am interested in the subgroup $G$ of $\textrm{...
Hans's user avatar
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Is this matrix adjugate's connection to combinatorial sequences known?

$\mathbf{SETUP}$ (Same setup as my previously posted question on determinants, but this time for the adjugate.) For my theoretical physics PhD I have been studying a model that requires the inversion ...
julianiacoponi's user avatar
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Is this matrix determinant's connection to the Partial Derangement / Rencontres numbers / subfactorial known?

$\mathbf{SETUP}$ For my theoretical physics PhD I have been studying a model that requires the inversion of an $n \times n$ matrix of this form: $$ \mathbf{A}_n= \begin{pmatrix} 1 & -a_{...
julianiacoponi's user avatar
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1 answer
72 views

Jacobi's formula via Determinant of Matrix Exponential, Equivalence of Equations?

Be $A$ a invertible and differentiable linear map. Jacobi's formula states that $\frac{d}{dt}\det{A} = det A \cdot tr(A^{-1}\cdot \frac{d\,A}{dt})$. It is a often derived corollary that $\det e^{B} = ...
theta_phi's user avatar
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Question on upper bound of rank of a matrix

I was working on this problem I found online (no solution available online). A and B are 2 matrices. Rank of A is 1 and rank of (A+B) is 3. What is the maximum possible rank of B? My progress: It is ...
LumosMaxima's user avatar
1 vote
1 answer
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Extension of restricted determinant

I'm looking for a function $f: (\mathbb{R}^{n \times n}, V \in \mathrm{Gr}(m, \mathbb{R}^n)) \rightarrow \mathbb{R}_{\geq 0}$ with the following properties: $\forall A, B \in \mathbb{R}^{n \times n},...
Jannis's user avatar
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Verification of a demonstration

I need to know if the proof I made for the following problem is correct. Problem: If C is a matrix of order $3 \times 3$ such that $\text{rank}(C) = 2$, then $\text{det}(C) = 0$ Proof: If it must be ...
studiare's user avatar
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Verify the result of a determinant (solved) [duplicate]

I need help with a problem where I am asked to verify this determinant: $$\begin{vmatrix} a & b & c & d \\ -b & a & d & -c \\ -c & -d & a & b \\ -d & c & -...
studiare's user avatar
1 vote
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Finding the Determinant of a Specific Tridiagonal Matrix

Given the matrix $A$: $$ A = \begin{pmatrix} a & b_1^* & 0 & \cdots & 0 & 0 \\ b_1 & a & b_2^* & \cdots & 0 & 0 \\ 0 & b_2 & a & \ddots & \vdots ...
Jaci Jean's user avatar
1 vote
1 answer
291 views

Jacobian of the vector reflection operator

While re-deriving some equations relevant to Monte-Carlo path tracing (specifically, the probability distribution of sampling a specific light direction from Sampling the GGX Distribution of Visible ...
lisyarus's user avatar
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1 vote
1 answer
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Does there exist a square matrix $B\neq O$ of order $n>1$, such that for every square matrix of order $n$ we have $\det(A+B)=\det(A)+\det(B)$?

This is a follow up to my previous question. There it was shown that for every square matrix $A$ of order $n$ there exists a square matrix $B\neq O$ of order $n$ with $\det(A+B)=\det(A)+\det(B)$. The ...
Hilbert's user avatar
  • 897
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Intuition for Laplace expansion

I've been trying to look for an intuitive understanding for the Laplace expansion of the determinant. I first tried looking for the proof but let's just say it was way to complicated for my ...
Ryan's user avatar
  • 127
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A variation of the Van der Monde determinant.

I have the following polynomial with $N$ variables $p(x_1,...,x_N) = \prod_{i=2}^{N} \left((i-1) x_{i} - \sum_{j=1}^{i-1}x_j\right)^{i-1}=(x_2-x_1)(2 x_3-x_2-x_1)^2...\left((N-1)x_N-x_{N-1}-...-x_1\...
Martin Daniel Jimenez's user avatar
1 vote
2 answers
116 views

Solve $\left|\begin{smallmatrix} 1 & a & a+x & a+x^2\\ a & 1 & a+x^2 & a+x\\ a+x & a+x^2 & 1 & a\\ a+x^2 & a+x & a & 1\\ \end{smallmatrix}\right|=0$

I have a problem that ask me about solving the equation $\begin{vmatrix} 1 & a & a+x & a+x^2\\ a & 1 & a+x^2 & a+x\\ a+x & a+x^2 & 1 & a\\ a+x^2 & a+x & a &...
MiguelCG's user avatar
  • 275
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0 answers
113 views

Determinant upper bound for specific matrix form

Let $A \in \mathbb{R}^{d \times d}$ be a symmetric matrix with the following properties: $A_{ij} \leq 0 \text{ for } i \neq j$ $A_{ii} \geq 0 \ \forall i \in [d]$ $\sum\limits_{i=1}^d A_{ij} = 0 \ \...
guysc's user avatar
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Determinant of a Vandermonde-like matrix with non-consequent powers

I need to calculate the determinant of a matrix which is built like a Vandermonde matrix, but has arbitrary increasing powers instead of consequent ones like $$ [M] = \begin{bmatrix} x_1^{n_1} & ...
Tigozawr's user avatar
1 vote
4 answers
128 views

Simplify $\begin {vmatrix} (x^2-a^2) & x^2-b^2 &x^2-c^2 \\\ (x-a)^3 & (x-b)^3 & (x-c)^3 \\\ (x+a)^3 & (x+b)^3 &(x+c)^3 \end{vmatrix}=0$

Given that $a,b,c$ are non-zero real and distinct constants, such that $$\begin {vmatrix} x^2-a^2 & x^2-b^2 &x^2-c^2 \\\ (x-a)^3 & (x-b)^3 & (x-c)^3 \\\ (x+a)^3 & (x+b)^3 &(x+...
mathophile's user avatar
  • 3,813
2 votes
2 answers
52 views

Let $X$ and $Y$ be two matrices of order $3\times 2$ and $2\times 3$. Then Least value of $|YX|$ is

Let $X$ and $Y$ be two matrices of order $3\times 2$ and $2\times 3$ respectively such that $XY=\begin{bmatrix} 2 & -2 & 0\\ -2 & 2 & 0\\ 0 & 0 &2\\ \end{bmatrix}$ Then ...
mathophile's user avatar
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$A,B \in GL_{n}$. Let $f(x) = det(xA+(1-x)B)$. Then conclude $f$ is a non constant polynomial.

The Actual Question $A,B \in GL_{n}$. Let $f(x) = det(xA+(1-x)B)$. Then conclude $f=0$ has finitely many solutions. Thoughts I understand that $f$ is a non zero polynomial, and if it's not a constant ...
Debu's user avatar
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1 vote
1 answer
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Given that a homogenous system of linear equation in three variable has a non trivial solution, evaluate the following expression.

The Question Given that the following homogeneous equation in three variable has a non trivial solution: \begin{equation*} \begin{array}{ccc} a x + y + z &=& 0 \\ x + b y + z &=& 0 ...
Debu's user avatar
  • 656
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0 answers
22 views

Determinant calculation and Lagrange interpolation polynomial

I've been trying to solve the following IMO-related question. Let $x_1,x_2,...,x_n$ be $n$ different reals, prove that $$\sum\limits_{i=1}^n\prod\limits_{\substack{j=1\\ j\neq i}}^n\frac{1-x_ix_j}{x_i-...
grj040803's user avatar
  • 198
2 votes
0 answers
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Second (and higher) derivatives of the adjugate operator for singular matrix

Consider the operator $$ f(A): A \rightarrow adj(A) $$ The derivative $Df_A (H)$ was given here and depends on the rank. I am interested in the second (and higher) derivatives in the $rank(A) = n-1$ ...
Luca Thiede's user avatar
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0 answers
24 views

Rule to simplify $\det{(X \: \text{diag}(\psi) \: X')}$ where X is not a square matrix?

Are there any known results/simplifications for an expression of the way $$\det{(X \: \text{diag}(\psi) \: X')} \:,$$ where X is a $n \times m$ matrix? I am aware that the determinant of a diagonal ...
mathmathmath's user avatar
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2 answers
34 views

A $\in$ $M_{4 \times 4}$. Let $A$ and $adjA$ (transpose of cofactor matrix of A) be non null matrix, and detA = 0. Find rank of A

The question: A $\in$ $M_{4 \times 4}$. Let $A$ and $adjA$ (transpose of cofactor matrix of A) be non null matrix, and detA = 0. Find rank of A. My attempt: I understand that the rank of A must be ...
Debu's user avatar
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0 votes
1 answer
37 views

Efficient Method to Calculate the Determinant

Given a matrix, the task is to find its eigenvectors. $$ \begin{pmatrix} 190 & 156 & -104 & -16 \\ -182 & -148 & 104 & 16 \\ 115.5 & 99 & -58 & -15 \\ -115.5 & ...
Gleb Cloudy's user avatar
0 votes
3 answers
69 views

calculating the determinant of a matrix with $1+a_i$ on the main diagonal [duplicate]

How should we verify the following determinant equality? $$ \left|\begin{array}{cccc} 1+a_{1} & a_{2} & \cdots & a_{n} \\ a_{1} & 1+a_{2} & \cdots & a_{n} \\ \vdots & \...
doctor's user avatar
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If A,B,C are three matrices of order $2$ , such that $|A+B|−|C|=−10 , |B+C|−|A|=25 , |C+A|−|B|=15$ then value of $|A+B+C|$ is equal to

If $A, B, C$ are three matrices of order $2$, such that $|A+B|-|C|=-10$, $|B+C|-|A|=25$, $|C+A|-|B|=15$ then value of $|A+B+C|$ is equal to ($|A|$ denotes the determinant of matrix $A$) My Approach: ...
mathophile's user avatar
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6 votes
0 answers
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Inequality involving a symmetric matrix and minors of an orthogonal matrix

Fix $n \geq 3$ and take any orthonormal vectors $x,y,z \in \mathbb{R}^n$. Let also $A \in M_n(\mathbb{R})$ be a symmetric matrix with positive entries ($A_{ij} = A_{ji} > 0$). Is the following ...
meler's user avatar
  • 175
1 vote
1 answer
101 views

Has this determinant already been calculated?

I've encountered the following gnarly looking determinant that I'm having trouble calculating: $$\begin{vmatrix} a_{k,1}&a_{k,2}&a_{k,3}&\cdots&a_{k,k-1}&b_k\\ a_{k-1,1}&a_{k-1,...
Vaskara_GRek_O's user avatar
1 vote
3 answers
173 views

Prove the formula for det(I + xA)

How do I show that the determinant and cofactors$(B_{kl})$ of matrix $B$ are given as follows? $$ B=I+xA=\left(\begin{array}{cccc} 1+a_{11}x & a_{12}x & \cdots & a_{1n}x \\\\ a_{21}x &...
arashi kyosuke's user avatar
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0 answers
45 views

How to Derive the Characteristic Polynomial of a Companion Matrix?

I am working on a problem involving the characteristic polynomial of a companion matrix and need some help understanding the derivation. Here is the matrix in question: $ C(p) = \begin{pmatrix} 0 &...
Herrpeter's user avatar
  • 1,326
2 votes
0 answers
66 views

Determinant of matrix is the discriminant

Let $f = t^n - \sum_{i = 0}^{n-1} a_i t^i \in k[t]$ a polynomial ($k$ a field). I construct a $2n \times 2n$ matrix $A$ in the following way: The upper-left quarter is the $n \times n$-identity ...
Verroq's user avatar
  • 98
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0 answers
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Linear relations between minors of a matrix

Let $X = (x_{ij})_{1 \leq i, j \leq n}$ by an $n \times n$-matrix over the field $Q = \mathbf{F}(x_{ij})_{ij}$ for some field $\mathbf{F}$. Let $[n] = \{1,\dotsc,n\}$. For subsets $I, J \subseteq [n]$,...
Bubaya's user avatar
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1 vote
1 answer
31 views

Submanifold of matrix space

One can identify the space $M_{\mathbb{R}}(n,n)$ of real $n \times n$ matrices as $\mathbb{R}^{n^2}$. Consider the subset $S:=\{ A \in M_{\mathbb{R}}(n,n) : det(A)=1 \}$ and show it is a smooth ...
Philip's user avatar
  • 523
1 vote
1 answer
25 views

An unusual archery contest scored by determinant

I had a dream last night where there was an archery contest held in three rounds with fixed duration, with three possible targets to aim for. The unusual thing about it was that the final score would ...
Daniel Schepler's user avatar
1 vote
0 answers
14 views

Results of invertibility of a matrix involving the Szego kernel

In the context of reproducing kernel Hilbert spaces, the Szego kernel is the function $k(z_i,z_j)=\frac{1}{1-z_j\overline{z_j}}$. Given two sets of points $\{z_1,\ldots,z_n\},\,\{w_1,\ldots,w_n\}\...
Math101's user avatar
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1 answer
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$\det(A^2-B^2) \leq \det(A^2)$ when $B$ is of rank 1

This is an exercise problem given at linear algebra class. As a problem before this I was able to show that $$\det(A + uv^T) = \det A + v^T (adj A )u$$ when $A$ is order $n$ square matrix and $u,v$ ...
mathhello's user avatar
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0 answers
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permutation of the determinant according to the groups $S_{n1}$ and $S_{n2}$?

Reading about alternating linear n maps, I found this alternative definition of a determinant, based on its permutation expression (which iteratively sums the product of all the permutations that can ...
MonkeyDL's user avatar
3 votes
1 answer
154 views

What $3 \times 3$ matrix gives the determinant $a^{2}+b^{2}+c^{2}$?

The $2D$ rotation matrix is given by a $2\times{2}$ matrix: $$\begin{bmatrix} a & -b\\ b & a \end{bmatrix}$$ The determinant of this matrix is $a^{2}+b^{2}$. Can one construct a $3\times{3}$ ...
Kyler Rusin's user avatar

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