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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.

2
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2answers
30 views

Higher dimensional cross product equivalent

I'm working on a computer vision script for high dimensions that is highly reliant on the cross product in 3D, but as far as I know, it is only formally defined in 3D and 7D. However, experimentally, ...
0
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0answers
8 views

Characterization of block matrices which only have positive principal minors (i.e which are P-matrices)

Questions Is there a characterization of block matrices $ S = \begin{pmatrix} A & B\\ B^T & D \end{pmatrix}$ which are P-matrices (i.e all principal minors are strictly positive) ? Is there a ...
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0answers
13 views

How to take the determinant of a rank(1,1) tensor?

I want to find the Jacobian matrix and its determinant of the generic infinitesimal transformation: $x'^\mu=x^\mu+\epsilon_\alpha\frac{\delta x^\mu}{\delta \epsilon_\alpha}$ where $\epsilon_\alpha$ ...
0
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0answers
40 views

The determinant after removing $2$ rows and columns.

Let $A_{ij}$ denote the $(i,j)$ cofactor. Let $A\bigl( \begin{smallmatrix} 1 & 2 & \cdots & (i-1) & (i+1) & \cdots & (j-1) & (j+1) & \cdots & n \\ 1 & 2 & ...
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0answers
15 views

Determinant of Hadamard product / sum of matrices (one diagonal)

I am trying to compute the determinant of $\boldsymbol{W}\odot \boldsymbol{S}$, where $\boldsymbol{S} \in PD(p)$ positive semidefinite matrix and $\boldsymbol{W}$ is a matrix whose diagonal entries $...
1
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1answer
36 views

How did they make this mathematical jump in steps, in finding a determinant of a simple 3x3 matrix?

Question I am looking at: Of the numbers 2, 3, and 5, which are eigenvalues of: [3 5 3] [1 7 3] [1 2 8] I know that we want to solve: ...
-1
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1answer
35 views

Matrix and determinant inequality

Let $A \in M_n {(\mathbb{R})} $ such that $A+A^T=2I_n$. Prove that $\det(A)\geq 1 $. I find out the form of $A$ and, respectively, its determinant, but i can not prove that it is bigger than 1, any ...
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0answers
33 views

Vandermonde Determinant with one column replaced [duplicate]

How to calculate the A(n) Vandermonde matrix determinant if the column with powers n-1 is replaced with powers n?
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0answers
33 views

Proof based on orthogonal matrix

I'm hoping to show that $|Q| = +1$ or $-1$ if $Q$ is a $p \times p$ orthogonal matrix. Since I know that $|QQ'| = |I|$ and $|Q||Q'| = |Q|^2$, then $|Q|^2 = |I|$ How should I approach this proof(...
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1answer
19 views

How do i generalize theory to arbitrary trace and arbitrary determinant?

Given a matrix $A\in \Bbb R^{2\times 2}$ Assume that trace $A = 0$. Then: a. If $\det A = 0$, then $0$ is the only eigenvalue. b. If $\det A <0$, then eigenvalue is $\pm\sqrt{-\det A}$ c. If $\det ...
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2answers
37 views

Limits and matrix determinants [on hold]

Let $A \in R_{3\times3}$ such that $\det(A)=\det(A+ \epsilon I_3)=0$, where $\epsilon=-\frac{1}{2} +\frac{i \sqrt{3}}{2} $. Find $$\lim_{n\to\infty}\frac{1}{n^4} \sum_{k=1}^{n}\det(A+k I_3)$$
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1answer
18 views

Determinant of large square matrix (term by term multiplication with same size matrix)

M is an N by N matrix with coefficients $a_{ij}$, B an N by N matrix with coefficients $b_{ij}$. Both M and B are symmetric matrices. I am trying to write the determinat of a matrix say, M .* B where ...
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3answers
33 views

What does determinant (of $3\times 3$ matrix) =0 mean?

Given three normalized vectors $$ \mathbf{u}=(\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3) $$ $$ \mathbf{v}=(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) $$ $$ \mathbf{w}=(\mathbf{w}_1, \mathbf{w}_2, \...
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1answer
31 views

Determinants Quadratic polynomials and iota in characteristic equation .

question: Given $A,B$ be two square matrices (with real entries )of order $2$ where $AB=BA$ $, \det(A)=\alpha>0$ , $\det(A+i\alpha B)=0$ then find value of L where , $L=\det(A^2-\alpha A B+...
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1answer
16 views

What does continuity of the determinant say about the value of a submatrix on neighborhoods of $M_{m\times n}(\mathbb{R})$

I am following Lee's book on smooth manifolds. On pages 19 and 20 he writes the following: Suppose $m < n$, and let $D_m \subset M_{m\times n}(\mathbb{R})$ be the set of real $m\times n$ ...
4
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1answer
83 views

Determinant of a $3\times 3$ matrix in simplest form.

Let $\alpha$ and $\beta $ be fixed non-zero reals and $f(n)=\alpha^n+\beta^n$ with $$A=\begin{pmatrix} 3&1+f(1)&1+f(2)\\1+f(1)&1+f(2)&1+f(3)\\ 1+f(2)&1+f(3)&1+f(4) \end{pmatrix}...
4
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1answer
59 views

Given $n^2$ different numbers to form a $n$ -degree matrix, prove that the number of possible determinants is at most $\frac{n^2!}{(n!)^2}$

Given $n^2$ different numbers from a field to form a $n$-degree matrix, prove that the determinant can take at most $$\frac{(n^2)!}{(n!)^2}$$ values. The number of different matrices is $(n^2)!$. ...
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1answer
47 views

Determinant with tangent functions

Proof:\begin{equation} \text{det} \begin{pmatrix} 1 & 1 & 1 \\ \tan A & \tan B & \tan C \\ \tan 2A & \tan 2B & \tan 2C \end{pmatrix}=0 \end{equation} ...
3
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2answers
175 views

Proving that two determinants are equal without expanding them

So I need to prove that $$ \begin{vmatrix} \sin^2(\alpha) & \cos(2\alpha) & \cos^2(\alpha) \\ \sin^2(\beta) & \cos(2\beta) & \cos^2(\beta) \\ \sin^2(\gamma) & \...
12
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3answers
795 views

Determinant of a particular matrix.

What is the best way to find determinant of the following matrix? $$A=\left(\begin{matrix} 1&ax&a^2+x^2\\1&ay&a^2+y^2\\ 1&az&a^2+z^2 \end{matrix}\right)$$ I thought it looks ...
0
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2answers
28 views

Is there any non zero matrix whose adjoint is a zero matrix

Just wanted to know whether their exits a non zero matrix whose adjoint is a zero matrix. And if so what would be inverse of a matrix whose adjoint as well as determinant is zero, as $$A^{-1}=\dfrac{1}...
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0answers
22 views

calculating deteminant of a 2*2 block matrix [closed]

Do you suggest a simple solve to calculate of the determinant of this block matrix? \begin{pmatrix} A & B \\ B^T & A \end{pmatrix} A and B are square matrices. A is symmetric matrix and B is ...
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0answers
27 views

symmetric matrix eigenvalues real (by induction)

Let's say we have an $n\times n$ symmetric matrix $A \in M_n(\Bbb R)$ and function $f(\lambda)=\det(A-\lambda I)$, where $I$ is the identity matrix. If we have some number $\lambda_0$ that is a ...
0
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2answers
28 views

understanding how an inverse determinant for 3*3 matrix is found

I’m trying to learn how to do a hill cipher encryption / decryption by hand. For decryption, I calculate the derterminant, but when I need to invert it, I don't understand how I can get it ... I ...
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2answers
78 views

Determinant of square matrices

This is my first question, please forgive me if I mistake something. The question I have is; Let A and B are square matrices of order 5. Find $\det(-A^3B^TA^{-1}) $ while $ \det(A)=5 $ and $ \det(...
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1answer
40 views

How to calculate determinant of matrix with 9's everywhere off-diagonal? [duplicate]

I need help about this question for my algebra exam. I would be very thankful if someone could help me solve this. I should calculate determinant of this $n \times n$-matrix: $$ \begin{pmatrix} ...
2
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1answer
31 views

Connection between ranks of an endomorphism and its linear image on the exterior power

Let $V$ be an $n$-dimensional real vector space, and let $1<k<n$. Let $\psi:\text{End}(V) \to \text{End}(\bigwedge^kV)$ be the exterior power map, $\psi(A)=\bigwedge^k A$. For $B \in \text{End}...
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3answers
42 views

How to calculate determinant of this matrix?

I need help about this question for my algebra exam. I would be very thankful if someone could help me solve this. I should calculate determinant of this matrix. $$ \begin{vmatrix} a & (a+...
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1answer
44 views

Given a $3\times 3$ matrix $A$, with $\det(A) = \frac{1}{8}$, find $\det(3A)$, $\det((6A)^{-1})$

How do I go about solving this? The $\det(3A)$ is not simply $3\cdot\frac{1}{8}$ correct?
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0answers
48 views

Divisibility of Determinant of matrix

Let $p$ be an odd prime number and $T_p$ be the following set of $2\times 2$ matrices $$ T_p= \biggl\{A = \begin{bmatrix}a & b\\c & a\end{bmatrix} \,\Big\vert\: a,b,c \in \{ 0, 1, 2, ... p-1\...
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3answers
45 views

Is it true that $\det(2*B)=160$, if $\det(B)=5$ [closed]

I want to clarify (because I can't find the answer) that I have to multiply diagonals, not multiply B by another B. B is a 5x5 matrix. So if I put ones and one 5 in middle , I have $2*2*2*2*10 = 160\\...
0
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3answers
55 views

For each $n\ge2$ calculate the determinant of this matrix

The matrix is $\left(\begin{matrix} 0 & 0 & \cdots&0&1 \\ 0 & 0 & \cdots&1&0 \\ \vdots&\vdots&\ddots & \vdots & \vdots\\ 0&1&\cdots&0&...
4
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2answers
97 views

Rank/determinant of the $n\times n$ matrix $((a_{ij}))$ where $a_{ij}=(i+j-1)^2$

I am trying to find the rank and determinant of the following $n\times n$ matrix : $$A=\begin{bmatrix}1^2&2^2&3^2&\cdots&n^2 \\ 2^2&3^2&4^2&\cdots&(n+1)^2 \\\...
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0answers
29 views

Proof for the determinant of a Cauchy Matrix

I want to proof the formula for the determinant of a Cauchy Matrix without recurring to matrix manipulation, but by directly applying the definition of the determinant. That is, given two sequences ...
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0answers
61 views

Determinant of a symmetric matrix with entries on diagonals

I am interested in the calculation of the determinant of the $N\times N$ symmetric matrix \begin{equation*} \mathbf B = \left(\begin{array}{*{20}c} 2 & & -1& &-1& &\\ & 2 &...
1
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3answers
44 views

Principal Minor Theorem for dimension 2

Let $A$ be a real ($2 \times 2$)-matrix such that the map $\mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}$, $\langle x,y \rangle = x^t A y$ is a scalar product. Now I would like to show that the ...
0
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2answers
31 views

Invertibility of a matrix.

Indicate why if a is a square matrix such that in each row and in each column one and only one element is non-zero then a is an invertible matrix. I tried to encompass the problem by determinants, ...
0
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2answers
30 views

Expected value of a $3 \times 3$ symmetric determinant with iid entries [closed]

We draw independently six numbers ($x_1,x_2,x_3,x_4,x_4,x_5,x_6$) from a same distribution with expected value $m$ and variance $\sigma ^2$. Next, we create a matrix $\begin{bmatrix} x_1&x_2&...
2
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0answers
47 views

Characteristic polynomial of a tree

I'm trying to understand the article: László Lovász, Jozsef Pelikan, On the Eigenvalues of Trees, Periodica Mathematica Hungarica, March 1973. I'm not sure I fully understand the proof of Lemma 1: if $...
2
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1answer
61 views

Proof that $A+A^T$ is singular

Let $A \in \mathbb R^{10,10}$, $x_{1},x_{2},...,x_{7}\in \mathbb R^{10}$ which are linearly independent vectors and $Ax_{1}=Ax_{2}=...=Ax_{7}$. Proof that $A+A^T$ is singular (not invertible) I am ...
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0answers
36 views

Compare ratio of two determinants before and after adding sum of rank 1 matrices in both determinants

I have a ratio between two determinants: $$r_1 = \frac{\det(\lambda I_d + \sum_{t=1}^{n}x_{t}{x_{t}}^T)}{\det(\lambda I_d)}$$ where $x \in \mathbb{R}^{d}$, $||x||_{2} \leq 1$, $\lambda > 0$, and $...
0
votes
1answer
35 views

matrices, determinant [duplicate]

i have the following homework problem: Find the determinant of X(s): X(s) = [s, 1, 1, 1], [1, s, 1, 1], [1, 1, s, 1], [1, 1, 1, s] I know i can exploit the fact that the product of the diagonal ...
1
vote
1answer
32 views

3x3 matrix operations intuition help

So I understand the intuition of taking the determinant of a 2x2 matrix, but what is the intuition for taking the determinant of 3x3, matrix? It makes zero intuitive sense just looking at it. Also, ...
2
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0answers
34 views

can we expand the relation between the determinant of a transformation and area of parallelogram to farther dimensions?

I'm now trying to understand determinants of linear transformations, and I was introduced to determinants as (the scaling factor that we multiply it by any area to figure out what the area of the ...
4
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1answer
64 views

Characterising minors of diagonal matrices

Let $k,d$ be positive integers, $1<k<d$. Let $\lambda_I=\lambda_{i_1,\ldots,i_k}$ be real numbers, indexed by multi-indices $I=(i_1,\ldots,i_k)$, where $1\le i_1<\ldots<i_k \le d$. Are ...
2
votes
1answer
29 views

Determinant of a matrix whose elements are trigonometric functions

Calculate: $$\det\begin{pmatrix} \cos\varphi & \sin\varphi & \cos\varphi & \sin\varphi \\ \cos2\varphi & \sin2\varphi & 2\cos2\varphi & 2\sin2\varphi \\ \cos3\varphi ...
1
vote
1answer
46 views

Determinant of a matrix with trigonometric functions

Prove: det $\begin{pmatrix} \cos(a-b) & \cos(b-c) & \cos(c-a) \\ \cos(a+b) & \cos(b+c) & \cos(c+a) \\ \sin(a+b) & \sin(b+c) & \sin(c+a) \end{pmatrix}=-2\sin(a-b)\...
0
votes
0answers
22 views

Is there a convenient way to compute the determinant of this anti-symmetric matrix? [duplicate]

$ \begin{pmatrix} 0 &a &b &c\\ -a &0 &d &e\\ -b &-d &0 &f\\ -c & -e& -f& 0\\ \end{pmatrix} $ It can be solved by Laplace Expansion, but I wonder if ...
1
vote
1answer
18 views

Why does this imply $F(k,n)$ is open?

Here, we have $F(k,n)$ defined as the set of ordered $k$-tuples of linearly independent vectors in $\mathbb{R^n}$. To start, let $X \in F(k,n)$. We can express $X$ as $$X = \begin{bmatrix} x_1^1 &...
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0answers
24 views

A special construction in $\mathbb{R}^n$

I have a question from my professor' notes. We defined $\Lambda^k(V)$ as the set of all $k$-Tensor' forms (multilinear transformations), $\omega$, which fulfuill $\omega(v_1,...,v_i,v_j,...,v_k)=-\...