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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3
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75 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 ...
0
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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{...
1
vote
1answer
86 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,...
0
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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 ...
-1
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1answer
47 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 &...
1
vote
2answers
66 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
votes
1answer
308 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
votes
1answer
210 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 ...
2
votes
0answers
78 views

The Maximum Singular Value of a Certain $\{1,-1\}$-Matrix

I have a $16$-dimensional real symmetric matrix with entries in $\{1,-1\}$. $11$ of the rows are pairwise orthogonal, so are the remaining $5$ rows. But the two orthogonal sets are not necessarily ...
2
votes
2answers
68 views

Calculate determinant of $ M=\left( \begin{array}{cc} A&-\vec d^T \\ \vec c& b \\ \end{array} \right) $.

I have a block matrix $$ M=\left( \begin{array}{cc} A&-\vec d^T \\ \vec c& b \\ \end{array} \right) $$ where $A$ is a $(n-1)\times (n-1)$ matrix, $\vec d,\ \vec c$ are two ...
3
votes
1answer
32 views

$n$ dimensional determinant using recurrence relations

Find determinant $$D_n(a,b,c)= \begin{vmatrix} a & b & 0 & 0 & \cdots & 0 & 0 & 0 \\ c & a & b & 0 & \cdots & 0 & 0 & 0 ...
1
vote
1answer
31 views

Calculating determinant of a matrix product

Let $M =\begin{pmatrix} 1 & 0 & ... & 0 \\ 0 & 1 & ...& 0 \\ ... & ... & ... & ...\\ 0 & 0 & 0 & 1 \\ x_1 & ...
2
votes
2answers
139 views

Use of determinants for vectors - what is the intuition behind it?

I am currently taking a Calculus 3 class. We just began using determinants in the study of vectors. I have some questions as to the apparent 'arbitrariness' of how we use determinants in the study of ...
-2
votes
3answers
93 views

Solve the determinant. [closed]

Prove that the following determinant is equal to $$2abc(a+b+c)^3$$ Using row column operations. $$ \det \begin{pmatrix} (b+c)^2 & a^2 & a^2 \\ b^2 & (c+a)^2 & ...
2
votes
3answers
228 views

How to prove $\det(I+uv^\intercal)=1+v^\intercal u$

Let be $u,v\in\mathbb{R}^n$, then $\det(I+uv^\intercal)=1+v^\intercal u $ where $I$ denotes the identity matrix of order $n$. How to prove this? what I did: let be $A=\{n\in\mathbb{N}: \...
2
votes
0answers
62 views

Maximal determinant of a $\{1,−1\}$ matrix of size $n$ is $2^{n−1}$ times the maximal determinant of a $ \{0,1\}$ matrix of size $n−1$.

Maximal determinant of a $\{1,−1\}$ matrix of size $n$ is $2^{n−1}$ times the maximal determinant of a $ \{0,1\}$ matrix of size $n−1$. How to prove this result? (I found this statement while ...
2
votes
1answer
31 views

Volume of the symmetric difference between a parallelotope and its translated.

Let $A$ be a n-dimensional parallelotope and $v \in \mathbb{R}^n$ a vector. Is there a formula giving the volume of the symmetric difference $A \Delta (v+A)$?
1
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3answers
91 views

Maximum determinant of $3 \times 3$ matrix

Good one guys! I'm studying to the maths olympiads in my college and I ran to the following problem: What is the possible matrix $3 \times 3$, that you can write using digits from $0 $ to $9$, (you ...
0
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0answers
22 views

Error on determinant from statistical errors on complex matrix elements

Say I have a complex matrix $A$ whose elements $A_{ij}$ have statistical error $\delta_{ij}$. I need to figure out from these errors what will be the error on the determinant $|A|$. If the matrix was ...
7
votes
3answers
135 views

If det $A = 0$ and $\det B \neq 0$ then show that $abc = -1$

This has been hurting my head for a while now.... If $$ \det\begin{bmatrix}a&a^2&1+a^3\\b&b^2&1+b^3\\c&c^2&1+c^3\end{bmatrix}=0 $$ And $$ \det\begin{bmatrix}a&a^2&1\\b&...
11
votes
3answers
234 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
120 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 $n \in \Bbb N \space \text{/{0}} \space \text{and} \space n_1,n_2 \in \Bbb N \space \text{such that} \space n_1+n_2=n$...
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4answers
56 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 ...
12
votes
4answers
1k 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\\ \end{...
0
votes
2answers
62 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
1answer
102 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
74 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 ...
1
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1answer
70 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
1answer
178 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 $||\det(A)|-|\...
1
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1answer
29 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
270 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
35 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 & 3x+...
4
votes
2answers
136 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
47 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 \\ ...
4
votes
1answer
67 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.
11
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1answer
153 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& a_{1}&...
1
vote
1answer
41 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
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0answers
97 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
62 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
48 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 = (...
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0answers
55 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 sub-...
2
votes
4answers
65 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
58 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
62 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
78 views

Parameter Matrix Determinant

$A=\begin{bmatrix} 7x+42&x-21&x-21&x-21&x-21\\x-21&7x+42&x-21&x-21&x-21\\x-21&x-21&7x+42&x-21&x-21\\x-21&x-21&x-21&7x+42&x-21\\x-21&...
6
votes
2answers
259 views

Determinant of a Certain Block Structured Positive Definite Matrix

PLEASE FIND THE EDITED VERSION OF THIS QUESTION HERE: Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence I WILL ALSO PUT UP A BOUNTY FOR THE EDITED VERSION....
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0answers
35 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
341 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 & 4\\...
2
votes
2answers
58 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
95 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 ...