Question about determinants, computation or 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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45 views

Algorithm for finding the value of determinant

Okay I am writing to write a program which computes the determinant of a matrix. So is there an algorithm that allows you to do that ? Are there any other ways of finding the determinant value other ...
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3answers
358 views

Why is determinant a multilinear function?

I am trying to understand (intuitive explanation will be fine) why determinant is a multilinear function and therefore to learn how elementary row operation affect the determinant. I understand that ...
2
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1answer
42 views

Alternating multilinear function satisfies $f(A)=\det(A)f(Id)$

I've just seen a proof of the statement: "Given $\alpha$ in a commutative ring $K$ there is a unique alternating multilinear function $f$ with $f(Id)=\alpha$." The determinant is defined as the ...
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1answer
42 views

Prove that det($A$) is non-zero iff $A$ is row equivalent to the $n\times n$ identity matrix

$A$ is an $n\times n$ matrix. Now if the row-reduced echelon form for this $A$ is $E$ then after all the row operations we have $\det(A)=M\det(E)$ where $M$ is a non-zero ...
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1answer
49 views

Integration of exponential matrix and determinant?

Is it possible to prove $$\int \exp\{-\frac{1}{2}(\beta-\hat\beta)^T(X^TH^{-1}X)(\beta-\hat\beta)\}\text{d}\beta=\{\det(X^TH^{-1}X)\}^{-1/2},$$ where $\hat\beta,X,H$ are all known? What additional ...
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1answer
64 views

What is $\mid\text{det}(A,G)\mid$?

I am reading an old paper dated back in 70', where I encounter this $$\mid\text{det}(A,G)\mid=(\text{det}\{(A,G)'(A,G)\})^{\frac{1}{2}}.$$ We compute the determinant of a single matrix, don't we? ...
5
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2answers
131 views

How to prove that this matrix is total unimodular

This matrix is total unimodular (tested by a computer program). ...
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1answer
84 views

relation between special linear group and special orthogonal group

What is the difference between special linear group and special orthogonal group ? The special linear group is the set of endomorphisms with determinant $1$. On the other hand, the special ...
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1answer
56 views

Find $a$ in the following matrix

I have the following question : matrix $A$ isn't diagonalizable while $a \in R$ $$A = \begin{pmatrix} 3 & 0 & 0 \\ 0 & a & a-2 \\ 0 & -2 & 0 \end{pmatrix}$$ Find $a$. I don'...
5
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4answers
154 views

Products of adjugate matrices

Let $S$ and $A$ be a symmetric and a skew-symmetric $n \times n$ matrix over $\mathbb{R}$, respectively. When calculating (numerically) the product $S^{-1} A S^{-1}$ I keep getting the factor $\det S$ ...
4
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2answers
82 views

Determinant of $ n \times n$ matrix and its characteristic polynomial.

Suppose, $M_4, M_5,..M_n$ is as follows then determinant and characteristic polynomial of $M_n$. $M_4=\left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 &...
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1answer
50 views

Finding the volume of the set of all $x \in \mathbb R^4$ satisfying $x^t A x \leq 1$ for a symmetric matrix $A$

Find the volume in $\mathbb R^4$ of the set of $x$ with $x^tAx \le 1$. You may use the fact that the volume in $\mathbb R^4$ of the set of $x$ with $|x|^2 = x^tx\le 1$ is $\frac{\pi^2}{2}$. My ...
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1answer
79 views

Convexity of Determinant of linear combination

Is it possible to show that the following is a convex function in $x$? $f(x)=\det(\sum_i x_i A_i)$ $A_i$ are real symmetric, positive definite matrices. Minkowski's inequality doesn't seem to do ...
2
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1answer
69 views

Why adjugate of $A$ is non singular, when $A$ is non singular?

Let $A$ be a non-singular square matrix. We know that $A \cdot \operatorname{adj}A = \det A \cdot I$. This implies that $\det\left(\operatorname{adj} A\right) = \left(\det A\right)^{n-1}$. Hence $\...
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0answers
37 views

Proof of determinant formula and coprime polynomial

Problem: Let $p(z)=p_o+p_1z+...+p_{n-1}z^{n-1}$ be a polynomial of maximum degree $n-1$. Show that $p(z)$ and $z^n-1$ are coprime if and only if $$\begin{vmatrix} p_0 & p_{n-1} & ... & ...
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59 views

Least squares have null determinant

I want fitting my data using bicubic interpolation: $$f(x,y)=\sum_{i=0}^{3}\sum_{j=0}^{3}a_{ij}x^iy^j$$ Let known $$f(0, 0)=1; f(2, 0)=1;f(1, 1)=0;f(0, 2) = 1; f(2, 2)=1$$ I used least squares method, ...
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136 views

Proving that $\det(A) = 0$ when the columns are linearly dependent

Proposition: Let $A$ be a $(n \times n)$-matrix. If the columns of $A$ are linearly dependent, then $\det(A) = 0$. Attempt at proof: Let $A = (A_1, A_2, \ldots, A_n)$, where each $A_i$ is a column ...
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0answers
60 views

Trace and Determinant of Field Extension

In algebra, we had a look at the trace and the determinant of a field extension. I am familiar with those concepts in linear algebra and I have seen that finite extensions can be viewed as a finite ...
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1answer
55 views

Prove or disprove: $|\det(Q)|=1 \Longrightarrow Q$ is unitary.

I wonder whether the statement of above can be written as an equivalence. So far I could prove the other direction $(\Longleftarrow)$: If $Q$ is unitary, then $1=\det(I)=\det(Q^HQ)=\det(Q^H)\det(Q)=...
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1answer
234 views

Proving wedge product is associative

Fix a real vector space $V$ of finite dimension. Let's denote by $\Lambda^p(V)$ the vector space of $p$-forms on $V$ (i.e. alternating $p$-tensors). Then we have the product $\wedge : \Lambda^p(V) \...
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2answers
90 views

Determinant of M [closed]

How to find the determinant of the $n\times n$ matrix $M$, whose all the entries are zero except 1st row, 1st column and diagonal entries: $$M= \begin{bmatrix} -x & a_2 & a_3 & \cdots &...
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0answers
34 views

Over what rings is the Hefferonian determinant unique?

Fix an $n\in\mathbb{N}$ and a field $\mathbb{K}$. A lot of texts in linear algebra like to define the determinant function on $\operatorname{M}_n\left(\mathbb{K}\right)$ as the unique function $\...
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3answers
596 views

Is this determinant always non-negative?

For any $(a_1,a_2,\cdots,a_n)\in\mathbb{R}^n$, a matrix $A$ is defined by $$A_{ij}=\frac1{1+|a_i-a_j|}$$ Is $\det(A)$ always non-negative? I did some numerical test and it seems to be true, but I've ...
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3answers
43 views

How can I show that and $n\times{n}$ matrix of the form in the description has a determinant of zero for $n>2$?

In General, $n>2$, $a_{i,j}=a_{i,j-1}+1$ and the matrix will be of the following form: $\begin{bmatrix}1&2&3&...&n\\n+1&n+2&n+3&...&n+n\\2n+1&2n+2&2n+3&...
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1answer
19 views

coefficient of $x$ in a determinant

What is the coefficient of $x$ in the expansion of the determinant$\begin{vmatrix} (1+x)^2 & (1+x)^4 & (1+x)^6 \\ (1+x)^3 & (1+x)^6 & (1+x)^9 \\ (1+x)^4 & (1+x)^8 & (1+x)...
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1answer
148 views

Determinant of 5x5 matrices

Let A and B be 5x5 matrices with det(-3A)=4 and det(B^-1)=2. Find the det(A), det(B) and det(AB). My answer : det(A)=-12 , det(B)=1/2 and det(AB)=-6. Wish to check my answer, thank you.
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1answer
66 views

Determinant of 3x3 matrices

Let $A$ and $B$ be $3\times3$ matrices with $\det(A)=10$ and $\det(B)=12$. Find $\det(AB)$, $\det(A^4)$, $\det(2B)$, $\det((AB)^T)$. Answers: $\det(AB)=\det(A)\det(B)=120$ , $\det(A^4)=10000$ , $\det(...
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1answer
42 views

What can be said about a matrix with a constant column of ones with entries from a finite field?

I am working with matrices of the following structure: $A = \begin{pmatrix} 1&\alpha_{21}&\cdots&\alpha_{n1}\\ 1&\alpha_{22}&\cdots&\alpha_{n2}\\ \cdots&\cdots&\ddots&...
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1answer
221 views

Bound on the difference of two determinants

Let $A$ and $B$ be two real, $n\times n$ matrices. Using Hadamard's inequality, it is not hard to show that $$ \left|\det A - \det B \right| \leq \|A-B\|_{2} \frac{\|A\|_{2}^n -\|B\|_{2}^n}{\|A\|_2 -\|...
2
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1answer
39 views

Determinant of a Certain 3 by 3 Block Matrix with Scaled Identity Blocks

What is the determinant or/and eigenvalues of the following 3 by 3 block matrix: $$\left[\begin{array}{ccc} \frac{3}{4}I & \frac{1}{4}I & \frac{1}{4}I \\ \frac{1}{4}I & \frac{3}{4}I & ...
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2answers
53 views

Technical question in Vandermonde determinat proof

I can follow the proof given in (2nd proof, or the induction proof), until the sentence: "From the Expansion Theorem for Determinants‎, we can see that the coefficient of $x_k$ is:". I don't ...
4
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2answers
98 views

Is this matrix invertible?

I have been working on a proof and am stuck with showing that the below matrix is invertible. I am not interested in the explicit inverse, only showing it has a nonzero determinant as the existence of ...
2
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1answer
80 views

How can I find $\det(A)/\det(B)$, when individual determinants blow up

I am interested in the quantity: $\frac{\det(A)}{\det(B)}$ of positive definite matrices $A$ and $B$. The problem I am running into now is that for large $A$,$B$, (around $200 \times 200$), the ...
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1answer
62 views

Linear Algebra: Question about determinants

The following matrices are $4 \times 4$ matrices. $$A=\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1\\ 1 & 1& 1 &0\\ 1 &1 &0 &0 \end{bmatrix}\\ B= \...
2
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0answers
146 views

Help me to prove the determinant formula

Actually it is about the question of n-linear function, but it is so relevant to the determinant formula. Here is the notation of the theorem. If $n>1$ and $A$ is an $n \times n$ matrix over $K$, ...
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2answers
38 views

Size of a triangle using determinant [duplicate]

find the size of a triangle using (determinant) with the following points: $(x_1,y_1)=(1,-2)$ $(x_2,y_2)=(-4,-2)$ $(x_3,y_3)=(-5,-1)$ How should I place those points in the determinant? ...
11
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2answers
309 views

Show that there is always a way to achieve det(A) > 0

a) Assume that $(a_1, ..., a_9)$ are different positive numbers. Let us make a $3 \times 3$ matrix $A_s$ by placing them arbitrarily into $9$ positions available. Show that there is always a way to ...
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1answer
142 views

Calculating determinant of matrix $n\times n$ [closed]

Given $$M := \mbox{diag} (1, 2, \dots, n) - n \, I_n + n \,1_n 1_n^T$$compute the determinant of $M$.
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1answer
95 views

Proving Things About Rings Using Things About Vector Spaces

All rings below are assumed to be commutative and having an identity. $\newcommand{\bw}{\bigwedge}\newcommand{\R}{\mathbf R}\newcommand{\mc}{\mathcal}$ Consider the following problem: Problem 1. ...
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4answers
41 views

Computing the eigenvalues of $\mathbb{1}-I$

Let $A=\mathbb{1}-I \in \{0,1\}^{n \times n}$, the matrix having 0 in the diagonal and 1 everywhere else. To compute the eigenvalues I tried to compute the characteristic polynomial using recursion, ...
2
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3answers
96 views

Can we infer that $\det(A+D)$ is always $\neq 0$, with $D$ diagonal matrix and $\det A=0$?

Let $A$ a $n\times n$ matrix, with $\det A=0$. Let $D$ a $n\times n$ diagonal matrix. Can we infer that $\det(A+D)$ is always $\neq 0$? I think the answer is "yes", but I do not know how to prove it.
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0answers
57 views

How to find the inverse of a matrix?

More specifically I mean, when we use row operations to find the inverse of matrix, we start by writing $(I\,|\, A)$, where $A$ is a matrix and $I$ is the identity matrix. But when we use column ...
3
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0answers
77 views

Proof of Leibniz formula from Laplace expansion

I'm trying to prove Leibniz formula for the determinant using Laplace expansion. Here's my attempt: For a $1 \times 1$ matrix $A = \begin{pmatrix}a_{11}\end{pmatrix}$, define $\det A = a_{11}$. For ...
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0answers
32 views

Find the value of a determinant in which the entries are in Harmonic Progression

Consider $9$ terms $a_1,a_2 \cdots a_9$ in Harmonic Progression with $a_4=5,a_5=4$. Find the value of the determinant $$\begin{vmatrix}a_1&a_2&a_3\\a_4&5&4\\ a_7&a_8&a_9\end{...
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1answer
88 views

How to prove this identity in vector calculus (suffix notation)?

Let $\epsilon_{ijk}$ be the alternating tensor defined by $$\epsilon_{ijk} = \begin{cases} 0, & \text{if any of $i$, $j$, $k$ are equal}\\ 1, & \text{if $(i,j,k)=(1,2,3)$, $(2,3,1)$ or $(3,1,...
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1answer
27 views

Measure of independency of vectors in a full rank matrix

Suppose A1 and A2 are two full rank matrices of similar size. What could be the parameter which say that one of matrix have more independent vectors compared to another matrix? In other words, column ...
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1answer
48 views

how to find inverse of a matrix

How to find the inverse of a 4x4 order matrix using adjoints for example $$A=\begin{pmatrix} 2 & -6 & -2 & -3 \\ 5 &-13 &-4 &-7 \\ -1 & 4& 1& 2 \\ 0 & 1 &...
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2answers
68 views

Formulate Cramer's rule for solving systems of linear equations, stating conditions under which the rule is applicable.

Formulate Cramer's rule for solving systems of linear equations, stating conditions under which the rule is applicable. Prove Cramer's rule for systems of two equations with two unknowns. So I just ...
0
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1answer
309 views

how to prove if $\det A=\det B$ then $A=CB$?

Let $A$ and $B$ be invertible $n \times n$- matrices and $C$ be an $n \times n$- matrix with $\det C =1$. Prove that $\det A = \det B$ if and only if $A=CB$. I've got the proof backward but I got ...
4
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1answer
217 views

What exacty is the role played by Jacobian or Wronskian?

In many of our derivations or in differential equations we come across the terms Jacobian or Wronskian. For example, to check the linear independence of solutions of differential equations, we ensure ...