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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10 views

Deriving a Formula for the determinant of a block matrix.

This is a follow up question to this. I want to solve the following problem: Let $n \in \Bbb N \space \text{/{0}} \space \text{and} \space n_1,n_2 \in \Bbb N \space \text{such that} \space ...
-4
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2answers
43 views

How many techniques are there to find the determinant of any NxN matrix? [on hold]

How many techniques are there in total to find the determinant of any NxN matrix? Please, only list the names. I found some:(1) Laplace's formula, (2) P.S. Listing only the names wouldn't make this ...
0
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1answer
45 views

What is the 2d equivalent of vector multiplication? [on hold]

If two three-dimensional vectors, v1 and v2, are multiplied (i.e. dot product), the result will be a 3x3 matrix. If, instead, there are two three-by-three matricies, what is the corresponding ...
1
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4answers
38 views

How to find a real matrix with complex eigenvalues,

Give a $2 \times 2$ real matrix $A$ with eigenvalues $2+3i$, $2-3i$. I would like hints only. So far, I've been trying get somewhere with $\det[A-(2+3i)I] = 0$ and $\det[A-(2-3i)I] = 0$; which ...
11
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4answers
503 views

Solving a system of non-linear equations

Let $$(\star)\begin{cases} \begin{vmatrix} x&y\\ z&x\\ \end{vmatrix}=1, \\ \begin{vmatrix} y&z\\ x&y\\ \end{vmatrix}=2, \\ \begin{vmatrix} z&x\\ y&z\\ ...
0
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2answers
29 views

Block Matrix Zero Determinant Implication?

Recently I've been working with a number of square (order of 2n) matrices whose determinants are zero. That is, $$\det\begin{bmatrix}A&B\\C &D \end{bmatrix} = 0$$ where each of A,B,C, and D ...
-5
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4answers
47 views

If $A$ is a $3\times3$ Matrics Then $\left |(2A)^{-1} \right |=?$ [on hold]

If $A$ is a $3\times3$ matrics.And $\left | A \right | = -7$.Then what's the value of $\left |(2A)^{-1} \right |$ Please help to do this math easily.I tried a lot but still no idea come into my ...
5
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1answer
79 views

Determinant of the Transpose of an Operator.

Let $V$ be a vector space over a field $F$ of characteristic $0$. A linear operator $T$ on $V$ induces a linear operator $\Lambda^k T:\Lambda^k V\to \Lambda^k V$ such that $\Lambda^k T(v_1\wedge ...
0
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1answer
19 views

Derivative of log determinant of triangular matrix

It is known that $$\frac{\partial\log|A|}{\partial A}=A^{-T}$$ However, if $L$ is a lower triangular positive definite matrix and take the log determinant, $\log |L|=\sum_i\log L_{ii}$. Question is ...
0
votes
1answer
42 views

Matrix with entries equal to $1$ and $-1$ (Sign Matrix)

What can we say about the determinant and (or) maximum eigenvalue of a matrix with entries equal to $1$ and $-1$. Further assume that the rows and columns are linearly independent. Are there special ...
6
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0answers
27 views

Lower bound on absolute value of determinant of sum of matrices

I needed to find a lower bound on $|\det(A+B)|$ where $|.|$ is the absolute value operator. Because I was unable to get such a bound so I was trying to guess a bound and prove it. But ...
1
vote
1answer
25 views

Calculating Determinant Using an Equation

$detA_{6x6} \neq 0$. $2A+7B=0$ Calculate $6det(2(A^t)^2B^{-1}A^{-1})$ My solution attempt: $A = -7/2*B$ and $det A^t = det A$ so $6det(2*A*(-7/2B)*B^{-1}A^{-1}) = 6det(-7)= 6*(-7)^6 = 705894$ ...
4
votes
2answers
89 views

Proof of determinant formula

I have just started to learn how to construct proofs. That is, I am not really good at it (yet). In this thread I will work through a problem from my Linear Algebra textbook. First i will give you my ...
0
votes
3answers
26 views

Number of real values of $x$ satisfying the following determinant?

The number of real values of $x$ which make the following determinant equal to $0$ are ? $$ \text{det}\left(\begin{matrix} x & 3x + 2 & 2x-1 \\ 2x-1 & 4x & ...
3
votes
2answers
48 views

Prove $\det(A - nI_n) = 0$.

Problem: Prove that $\det(A - n I_n) = 0$ when $A$ is the $(n \times n)$-matrix with all components equal to $1$. Attempt at solution: I tried to use Laplace expansion but that didn't work. I see the ...
0
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1answer
36 views

Calculation of determinant using its properties [duplicate]

The task is to calculate the following determinant by using the properties of a determinant: $$\begin{vmatrix} n^2 & (n+1)^2 & (n+2)^2 \\ (n+1)^2 & (n+2)^2 & (n+3)^2 \\ ...
3
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1answer
41 views

Prove that $\det\left[A^{T}B-B^{T}A\right]=\det[A+B]\cdot\det\left[A-B\right]$

So I need to prove that: $$\det\left[A^{T}B-B^{T}A\right]=\det[A+B]\cdot\det\left[A-B\right]$$ where $A$, $B$ are two orthogonal matrices, but it seems I'm missing something.
10
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1answer
120 views

Determinant of a special $4\times 4$ matrix

Let $f(x)=\sum_{k=1}^{4}a_{k}x^{k},\varepsilon =\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}.$ $\qquad\qquad 4\times 4$ matrix $$T=\begin{bmatrix} 1& a_{2}& a_{3}& a_{4}\\ 1& ...
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0answers
9 views

Find inverse and determinant of a symmetric matrix - for a maximum-likelihood estimation

Evaluate the determinant $\det \Omega $ and find the inverse matrix $\Omega^{-1}$ of: $$\Omega = \begin{bmatrix} \beta_1^2(1+\theta_1^2) & \beta_1 \beta_2 & ... & \beta_1 \beta_{k-1} ...
1
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1answer
34 views

Product of $A$ with the adjoint of $A$: why are all nondiagonal elements zero?

Let \begin{align*} A = \begin{pmatrix} 1 & 2 & 4 \\ 3 & 2 & 1 \\ 6 & 8 & 2 \end{pmatrix}. \end{align*} We have $\det(A) = 44$. The cofactor matrix corresponding with $A$ is ...
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0answers
20 views

Proving determinant of Vandermonde matrix

Problem: A matrix of the form \begin{align*} A= \begin{pmatrix} 1 & x_1 & x_1^2 & \cdots & x_1^{n-1} \\ 1 & x_2 & x_2^2 & \cdots & x_2^{n-1} \\ \vdots \\ 1 & x_n ...
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0answers
27 views

Why is the computation of the Jacobian determinant different for certain integrations?

I am used to computing the Jacobian, when, say, changing from x,y coordinates to u,v coordinates, as computing the determinant of the derivative matrix of $x_u$,$x_v$, $y_u$, $y_v$, i.e., ...
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0answers
38 views

Need clarification regarding a proof about the determinant of a block matrix

Let $A= (a_{ij}) \in M_n (F)$ be of the form \begin{align*} A = \begin{pmatrix} B & C \\ O & D \end{pmatrix}, \end{align*} where $B = (b_{ij}) \in M_r (F), D = (d_{ij}) \in M_s (F)$ and $C = ...
1
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0answers
20 views

Invertible matrices, permutations and leading principal minors

Given an invertible $\{-1,0,1\}$-matrix $A$ (its determinant is $\pm 1$), are there two permutation matrices $P$ and $Q$ such that all the leading principal minors (determinants of the top-left ...
2
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4answers
50 views

A determinant made of $n \times n$ determinants.

I came across this problem, in a recent exam. So I was given three matrices $$ A, B, C \in M_{n} (\mathbb{R}) $$ and that $$ 0 \in M_{n}(\mathbb{R}) $$ is the zero matrix. Then I was also given the ...
1
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1answer
41 views

Determinants and monic polynomials [duplicate]

I wish to show that $$ \det \begin{pmatrix} x & a & a & a\\ a & x & a & a\\ a & a & x & a\\ a & a & a & x \end{pmatrix}=(x-a)^3(x+3a).$$ Obviously, I ...
2
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0answers
49 views

Can I perform elementary line operations to $\det(A-\lambda I)$ like this?

I have a $4 \times 4$ matrix: $$A = \begin{pmatrix} 2 & 3 & 1 & 0 \\ 4 & -2 & 0 & -3\\ 8 & -1 & 2 & 1\\ 1 & 0 & 3 ...
2
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4answers
55 views

Parameter Matrix Determinant

$A=\begin{bmatrix} ...
5
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1answer
102 views
+50

Determinant of a Certain Block Structured Positive Definite Matrix

Is there a lower bound for the determinant or minimum eigenvalue of the following $d$ by $d$ matrix in terms of $d$? $$\Gamma=\left( {\begin{array}{cc} I & B \\ B^{*} & I \\ \end{array} ...
2
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0answers
31 views

Show $\int_{\mathbb{R}^n} \exp(-\|Ax\|^2) d\mu(x)=\frac{\pi^{\frac{n}{2}}}{|\det(A)|}$

$A$ is a invertible $n \times n$ matrix. Show: $\int_{\mathbb{R}^n} \exp(-\|Ax\|^2) d\mu(x)=\frac{\pi^{\frac{n}{2}}}{|\det(A)|}$ Can someone give me a hint on how to show that?
2
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1answer
52 views

Easiest way to calculate the determinant of this 4x4 matrix

I have this 4x4 matrix: $$A= \begin{pmatrix} 2 & 3 & 1 & 0 \\ 4 & -2 & 0 & -3\\ 8 & -1 & 2 & 1\\ 1 & 0 & 3 & ...
2
votes
2answers
53 views

Find the jacobian

I'm been struggling with the problem for a quite some time now. I need to find the jacobian for the following : $$u=x-y$$ $$v=xy$$ What I did : $$x=y+u\\x=\frac{v}{y}\\y=x-u\\y=\frac{v}{x}$$ ...
1
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0answers
33 views

An identity with determinant and trace of a matrix

How to prove the following identity: $$\det(A)=\frac{1}{d!}\sum_{\sigma\in S_d}\mathrm{sgn}(\sigma)\mathrm{Tr}_{\sigma}(A)$$ where $\mathrm{Tr}_{\sigma}(A)$ is defined as following if $\sigma$ is ...
0
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1answer
43 views

Determinant of an $n \times n$ matrix,problem

$ A_=\begin{pmatrix} ...
0
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0answers
28 views

Show a regular complex symmetric square matrix is reversible

Find following complex symmetric square matrix's canonical form under the congruent ( through complex square matrix ),where $i^2=-1$. $$\sum\limits_{1 \le k < l \le n} {\left( {k + il} ...
0
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1answer
23 views

How to check the determinant and rank of multiplied matrices?

Given $A \in\mathbb{R^{7\times8}}$, $B \in\mathbb{R^{8\times5}}$ and $C \in\mathbb{R^{5\times7}}$ How can one check whether $$det(ABC) = 0$$ is true? Given their spaces, the multiplications are ...
2
votes
2answers
29 views

Matrix roots of the characteristic equation

Let A be a matrix of $n \times n$ dimensions and $p( \lambda)= \det (A- \lambda I)$. Then $p(A)=0$ by Caylee-Hamilton. Are there any other matrices that satisfy the characteristic equation of A?
2
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0answers
27 views

Geometric Interpretation of Determinant of Transpose

Below are two well-known statements regarding the determinant function: When $A$ is a square matrix, $\det(A)$ is the signed volume of the parallelepiped whose edges are columns of $A$. When $A$ is ...
1
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1answer
29 views

Using a determinant to find the Cartesian equation for a plane from its parametric equations

This horribly unreadable webpage describes a method to find the Cartesian equation for a plane given its parametric equations. I'll try to type the method out here in a neater fashion: The ...
0
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2answers
35 views

Signing of a binary matrix to a totally unimodular matrix

I have the following binary matrix: \begin{pmatrix} 1& 1& 1& 0 \\ 0& 1& 1& 1\\ 1& 0& 1& 1\\ 1& 1& 0& 1\\ \end{pmatrix} Definition: Signing a matrix ...
1
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0answers
53 views

Find the Determinant Question

Find the determinant $$ \begin{vmatrix} \dfrac1{a_1+b_1} & \dfrac1{a_1+b_2} & \ldots & \dfrac1{a_1+b_n} \\ \dfrac1{a_2+b_1} & \dfrac1{a_2+b_2} & \ldots & \dfrac1{a_2+b_n} ...
0
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0answers
25 views

Why does this equality stand?

We have that $$\frac{\partial}{\partial{t}}J=\begin{vmatrix} \frac{\partial}{\partial{t}}\frac{\partial{\xi}}{\partial{x}}& \frac{\partial{\eta}}{\partial{x}} & ...
2
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2answers
54 views

If $(I-A)(I+A)^{-1}$ is orthogonal then prove that A is skew symmetric.

Question from Determinants.Can't solve !
6
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2answers
72 views

Order $n^2$ different reals, such that they form a $\mathbb{R^n}$ basis

I've been trying to solve this linear algebra problem: You are given $n^2 > 1$ pairwise different real numbers. Show that it's always possible to construct with them a basis for $\mathbb{R^n}$. ...
0
votes
1answer
19 views

Computing determinant of the matrix $C$

Let $$C=\begin{bmatrix} 0 & 0 & \cdots &0 & -c_0 \\ 1 & 0 & \cdots & 0& -c_1 \\ 0& 1 & \cdots & 0& -c_2 \\ \vdots & \vdots & & & \\ 0 ...
3
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4answers
61 views

Show that $A$ and $A^T$ do not have the same eigenvectors in general

I understood that $A$ and $A^T$ have the same eigenvalues, since $$\det(A - \lambda I)= \det(A^T - \lambda I) = \det(A - \lambda I)^T$$ The problem is to show that $A$ and $A^T$ do not have the same ...
1
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1answer
31 views

Determinants using elementary row operations

Let matrix $A$ be defined as \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ ...
3
votes
2answers
99 views

Show determinant of $\left[\begin{matrix} A & 0 \\ C & D\end{matrix}\right] = \det{A}\cdot \det{D}$

Let $A \in \mathbb{R}^{n, n}$, $B \in \mathbb{R}^{n, m}$, $C \in \mathbb{R}^{m, n}$ and $D \in \mathbb{R}^{m, m}$ be matrices. Now, I have seen on Wikipedia the explanation of why determinant of ...
9
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1answer
144 views

Show that a matrix has positive determinant

For a natural number $i>0$, let $p_i$ be the $i$th prime number, that is, $p_1=2, p_2=3, p_3=5,...$. Show that for all $n$, the following matrix has positive determinant $$ \begin{pmatrix} ...
3
votes
2answers
238 views

Determinant of matrix with trigonometric functions

Find the determinant of the following matrix: $$\begin{pmatrix}\cos\left(a_{1}-b_{1}\right) & \cos\left(a_{1}-b_{2}\right) & \cos\left(a_{1}-b_{3}\right)\\ \cos\left(a_{2}-b_{1}\right) ...