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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How to prove $\det{AB} = \det{BA} = \det{A}\det{B}$? [duplicate]

Possible Duplicate: How to show $\det(AB) =\det(A)\det(B)$ How would I prove that $$\det{AB} = \det{BA} = \det{A}\det{B}$$
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1answer
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problem on determinant as a linear map

For $V = (V_1,V_2 )\in\mathbb{R}^2$ and $W = (W_1,W_2 )\in\mathbb{R}^2$ , Consider the determinant map $$\det :\mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}$$ defined by $$\det(V,W) = V_1W_2 ...
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1answer
678 views

finding values for determinant to equal 0

I needed to find for which values of $\lambda$ the matrix is singular. $$ \begin{bmatrix} 1-\lambda & 0 & 3 \\ 1 & 1-\lambda & 0 \\ 0 & 2 & ...
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2answers
874 views

Determinant of a big matrix

I have done a program to calculate a determinant of a matrix. My program works, but the problem is that it takes long time to calculate, especially for big matrix. Could you tell me how can a perform ...
2
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0answers
91 views

Determinant, number of non zero columns

Trying to build a reduction from the maximum coverage problem to my research problem, I'm facing this difficulty : Let $X$ be a $n \times m$ binary matrix (with $m > n$), can we define a square ...
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2answers
117 views

a question about how to calculate a determinant [duplicate]

Possible Duplicate: Determinant of a specially structured matrix Determining eigenvalues, eigenvectors of $A\in \mathbb{R}^{n\times n}(n\geq 2)$. I have the following matrix $$ A = ...
2
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5answers
134 views

Determinant with unknown parameter.

I'm given 4 vectors: $u_1, u_2, u_3$ and $u_4$. I'm going to type them in as points, because it will be easier to read, but think as them as column vectors. $$u_1 =( 5, λ, λ, λ), \hspace{10pt} u_2 ...
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1answer
628 views

Volume of a pyramid as a determinant?

I have three given points, A, B and C, each of them is a corner of a pyramid. Another corner is located in origo. The task is to set up a determinant to describe the pyramids volume. ...
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2answers
108 views

How to show a Determinantal inequality

If $A, B$ and $C$ are $n\times n$ positive semidefinite matrices. How to show that $$\det(A + B) + \det(A + C)\le \det A + \det(A + B + C)?$$
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4answers
137 views

Show that for vectors $\bf u$ and $\bf v$ in $ℝ^3$, $\bf u \times v = (-v) \times u$

How would I use the properties of determinants to show that for any two vectors $\bf u$ and $\bf v$ in $ℝ^3$ $$\bf u \times v = (-v) \times u$$
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2answers
159 views

Finding a simple expression for this determinant

Evaluate $$\Delta=\left\lvert\matrix{ 1 & x & yz \\ 1 & y & zx \\ 1 & z & xy }\right\rvert$$ The answer of the above question is $(x-y)(y-z)(z-x)$. But will solving I ...
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2answers
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By using properties of determinants show that

$$\begin{vmatrix}1+a^2-b^2&2ab&-2b\\ 2ab&1-a^2+b^2&2a\\ 2b&-2a&1-a^2-b^2\end{vmatrix}=(1+a^2+b^2)^3$$ I have been trying to solve the above determinant. But unfortunately my ...
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1answer
49 views

A linear algebra problem - matrix equation

Let $\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_n$ column vectors, each with the same $n$ components. So: \begin{equation} \mathbf{v}_i = \left[\begin{array}{c}v_i\\ v_i \\\vdots \\ ...
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2answers
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What is the origin of the determinant in linear algebra?

We often learn in a standard linear algebra course that a determinant is a number associated with a square matrix. We can define the determinant also by saying that it is the sum of all the possible ...
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1answer
617 views

Determinant of a symmetric matrix

Given an $n\times n$ matrix $C= [c_{ij}]$ which is symmetric (i.e. $c_{ij}=c_{ji}\ \forall i,j$) calculate the determinant of the following matrix (assume $c_{ij} \neq 0\ \forall i,j$): ...
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1answer
156 views

determinant of an $ n\times n$ matrix type [duplicate]

Possible Duplicate: How to calculate the following determinants Computing determinant of a specific matrix. How can one compute the determinant of an $n\times n$ matrix where all the ...
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1answer
114 views

Determinant of matrices along a line between two given matrices

The question, with no simplifications or motivation: Let $A$ and $B$ be square matrices of the same size (with real or complex coefficients). What is the most reasonable formula one can find for ...
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1answer
70 views

Solve a so that rank of Matrix A won't be full

Solve for $$ \alpha\epsilon\mathbb{R} $$ so that rank of matrix $A$ won't be full. Find all results of the system $Ax = b$, where $$ b = [2\hspace{2 mm} 6\hspace{2 mm} 7]^T $$ $$A= \left( ...
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0answers
351 views

Determinant of Transpose of Linear Map

I'm trying to find a way to prove that the determinant of the transpose of an endomorphism is the determinant of the original linear map (i.e. det(A) = det(Aᵀ) in matrix language) using Dieudonne's ...
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1answer
97 views

Evaluating this determinant

I am asked to find the following determinant $$D = \begin{vmatrix} 1 & 2 & \cdots & n \\\ n+1 & n+2 & \cdots & 2n \\\ \vdots & \vdots & \vdots & \vdots \\\ ...
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1answer
397 views

collinear three points in a linear equation $ax+by+z$

So, there is a plane equation $ax+by+z = 0$. Suppose there are three points $(x_1,y_1,z_1)$, $(x_2,y_2,z_2)$, $(x_3,y_3,z_3)$ that is on the given plane. Then, $\begin{vmatrix} x & y & z ...
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2answers
207 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 ...
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0answers
169 views

Intuition in permutations for Laplace Determinant Expansion

Starting with the Leibniz formula for the determinant, I wish to derive the Laplace (Cofactor) Expansion. At the risk of being overly verbose, please see the proof here. Now I understand the idea of ...
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1answer
343 views

Laplace expansion

This statement is from the book of Winitzki Linear Algebra via Exterior Products. (Section 3.4, page 123) Let $V$ be finite dimensional vector space, $\dim(V)=N$. The determinant of the matrix ...
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5answers
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Find the determinant of $I+A$

Let $A$ be a $2\times2$ matrix with real entries such that $A^2=0$.Find the determinant of $I+A$ where $I$ denotes the identity matrix. I proceed in this way :Note that $(I+A)A=A+A^2 \Longrightarrow ...
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1answer
484 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 ...
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2answers
76 views

What is the correct way to solve $|2K^3-2K^4|$ determinant?

Given - $$K_{3\times3} = \begin{bmatrix} 1&1&1 \\ 3&2&1 \\ 1&2&1 \end{bmatrix}$$ $$|K| = 2$$ Find - $$|2K^3-2K^4|$$ I tried this: Since $|A+B|=|A|+|B|$ ( ...
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1answer
96 views

Simpler expression for a certain determinant.

A question in elementary linear algebra, while considering the Cayley-Menger Determinant: Given an $n\times n$ matrix $M$, consider $$\tilde{M}=\begin{pmatrix} M & (1,1,\cdots, 1)^\top \\ ...
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5answers
927 views

Suppose A is an n-by-n matrix with its diagonal entries are n and other entries are one. Find determinant of A.

For $n \geq 2$, find the determinant of $A_{n}=\begin{bmatrix} n & 1 & 1 &\ldots &1 \\ 1 & n & 1 &\ldots &1 \\ 1 & 1 & n &\ldots &1 \\ \vdots & ...
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1answer
112 views

Does the identity $\det(I+g^{-1})\det(I+g)=|\det(g-I)|^2$ hold for $g \in U(n)$?

In a paper (corollary 1, p.14) the following identity is used: Let g be a unitary matrix. Then: $$\det(I+g^{-1})\det(I+g)=|\det(g-I)|^2 \text{ for }g \in U(n)$$ Now my question is why this ...
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3answers
929 views

Factorise the determinant $\det\Bigl(\begin{smallmatrix} a^3+a^2 & a & 1 \\ b^3+b^2 & b & 1 \\ c^3+c^2 & c &1\end{smallmatrix}\Bigr)$

Factorise the determinant $\det\begin{pmatrix} a^3+a^2 & a & 1 \\ b^3+b^2 & b & 1 \\ c^3+c^2 & c &1\end{pmatrix}$. My textbook only provides two simple examples. Really have ...
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0answers
113 views

matrix construction

Given any matrix $A$, can one construct a matrix $B$ such that $B$ is nonnegative and the spectral radius of $B$ is strictly less than 1 the determinant of $A$ is equal to the first entry of $B^*$ ...
2
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2answers
96 views

Bijection from $S_{n-1}$ to $\{\sigma \in S_{n} : \sigma(k) = j \}$

Let $n$ be a natural number. Let $k$ be an element of $\{1, \ldots , n\}$. For each j in $\{1, \ldots , n\}$, I want to find a bijection $f_j$ from $S_{n-1}$ to $\{\sigma \in S_n : \sigma(k) = j ...
5
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0answers
240 views

Symmetric functions of the eigenvalues of A+B, A, B, ABA, BAB, et.c.

(this is an improved version of What about other symmetric functions of the eigenvalues? ) Let $A$ be a matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Then $\det(A) = \lambda_1 \dots ...
7
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2answers
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Determinant of an $n\times n$ complex matrix as an $2n\times 2n$ real determinant

If $A$ is an $n\times n$ complex matrix. Is it possible to write $\vert \det A\vert^2$ as a $2n\times 2n$ matrix with blocks containing the real and imaginary parts of $A$? I remember seeing such a ...
2
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2answers
214 views

Computing determinant of a specific matrix.

How to calculate the determinant of $$ A=(a_{i,j})_{n \times n}=\left( \begin{array}{ccccc} a&b&b& \cdots & b\\ b& a& b& \cdots& b\\ \vdots& \vdots& \vdots& ...
6
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5answers
326 views

How to prove $I + t X$ is invertiable for small enough $ | t | ?$

Let $X \in \text{GL}_n(\mathbb{R})$ be an arbitrary real $n\times n$ matrix. How can we prove rigorously: $$ \underset{b>0} {\exists} : \underset{|t|\le b} {\forall} : \det (I + t X) \neq 0 $$ If ...
11
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2answers
401 views

Integral of determinant

Good evening. I need help with this task $$ \int\limits_{-\pi}^\pi\int\limits_{-\pi}^\pi\int\limits_{-\pi}^\pi{\det}^2\begin{Vmatrix}\sin \alpha x&\sin \alpha y&\sin \alpha z\\\sin \beta ...
3
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1answer
92 views

Characteristic equation for 2-nd order ODE

Given a differential equation $\dot x = Ax$, $x \in \mathbb{R}^n$ we define its characteristic equation as $\chi(\lambda) = \det (\lambda I - A)$. Consider now the second order ODE $$ \ddot x + A x ...
3
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5answers
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calculate generally the determinant of $A = a_{ij} = \begin{cases}a & i \neq j \\ 1 & i=j \end{cases}$

calculate generally the determinant of $A = a_{ij} = \begin{cases}a & i \neq j \\ 1 & i=j \end{cases} = \begin{pmatrix} 1 & a & a & · & a \\ · & · & · & · \\ a ...
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7answers
357 views

positive definite quadratic form

Is $\sum_{i=1}^n x_i^2 + \sum_{1\leq i < j \leq n} x_{i}x_j$ positive definite? Approach: The matrix of this quadratic form can be derived to be the following $$M := \begin{pmatrix} 1 & ...
7
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1answer
151 views

How to show that $\mathrm{SL}(2,\mathbb Z) = \langle A, B\rangle$?

Show, that if $\mathbf{A}= \left( \begin{array}{cc} 1&1\\ 0&1 \end{array} \right)$, $\mathbf{B}= \left( \begin{array}{cc} 0&1\\ -1&0 \end{array} \right)$ and $\mathrm{SL}(2, ...
5
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1answer
219 views

A particular (functional) determinant calculation

One wants to calculate the quantity, $\det'(\frac{\partial}{\partial t} - i [\alpha, ])$ where the prime on the "det" means that one wants to do a product over only non-zero eigenvalues of the ...
6
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1answer
232 views

Do multiplicative maps of matrices factor through determinants?

Given a map $f:M_n(k)\to k$ (with $k$ some field) such that $f(AB)=f(A)f(B)$ for all matrices $A$ and $B$, is it necessarily the case that $f$ factors through the determinant, i.e. does there exist a ...
3
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3answers
242 views

Rank and determinant of $D$ , an $n\times n$ real matrix, $n\ge 2$

Let $D$ be a $n\times n$ real matrix, $n\ge 2$. Which of the following is valid? $\det(D)=0\Rightarrow \mathrm{rank}(D)=0$ $\det(D)=1\Rightarrow \mathrm{rank}(D)\neq 1$ $\det(D)=1\Rightarrow ...
6
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2answers
272 views

Determinant called Grammian

Famously, if functions $f_1,f_2,…,f_n$, each of which possesses a derivative of order $n-1$, are linearly independent on the interval $I$, if $$ \det\left( \begin{array}{ccccc} f_1 & f_2 & ...
56
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3answers
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Cute Determinant Question

I stumbled across the following problem and found it cute. Problem: We are given that $19$ divides $23028$, $31882$, $86469$, $6327$, and $61902$. Show that $19$ divides the following determinant: ...
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1answer
168 views

Cross product determinant's matrix

The cross product $a \times b$ can be represented by the determinant $$\mathbf{a}\times\mathbf{b}= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & ...
1
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1answer
63 views

Determinant after matrix change issue

So, the problem is this. I can infer that the determinant of a matrix with two identical rows is equal to $0$ because exchanging two rows negates the determinant of the matrix (which is relatively ...
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1answer
128 views

2-by-2 matrix determinant subrtraction

How to calculate determinant of this matrix? $\left[\begin{array}{cc} 1 & 2 \\ 0 & -2 \\ \end{array}\right]^3 . \left[\begin{array}{cc} 2 & 3 \\ -1 & 1 \\ ...