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
votes
3answers
170 views

Find this Determinant

I have to find this determinant, call it $D$ \begin{vmatrix} \frac12 & \frac1{3}& \frac1{4} & \dots & \frac1{n+1} \\ \frac1{3} & \frac14 & \frac15 & \dots & ...
0
votes
2answers
23 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
votes
2answers
54 views

Crout matrix decomposition [on hold]

In naive terms and step by step, how to to find the determinant of any NxN matrix by using LU Decomposition of Crout's method. Also, discuss its efficiency as compared to other LU decomposition ...
0
votes
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
vote
4answers
43 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
votes
4answers
508 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
votes
2answers
30 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
votes
4answers
47 views

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

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
votes
1answer
80 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
votes
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
47 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
votes
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
90 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
votes
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
votes
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
votes
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& ...
0
votes
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
vote
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 ...
0
votes
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 ...
1
vote
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., ...
1
vote
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
vote
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
votes
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
vote
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
votes
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
votes
4answers
55 views

Parameter Matrix Determinant

$A=\begin{bmatrix} ...
5
votes
2answers
119 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
votes
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
votes
1answer
53 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
vote
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
votes
1answer
43 views

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

$ A_=\begin{pmatrix} ...
0
votes
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
votes
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
30 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
votes
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
vote
1answer
30 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
votes
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
vote
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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
1answer
146 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} ...