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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14
votes
1answer
167 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
votes
1answer
25 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 & ...
1
vote
2answers
41 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
votes
2answers
80 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
votes
1answer
69 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 ...
0
votes
1answer
57 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
votes
0answers
66 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 $K$, ...
0
votes
2answers
28 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 ...
11
votes
2answers
271 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 ...
0
votes
0answers
33 views

Rank-one update Determinant

$I_{MN}$ is an $(M*N,M*N)$ real matrix. $S$ is an $(N,M)$ real matrix. $S^T$ is an $(M,N)$ real matrix. $|A|$ stands for $\det(A)$. $k,w$ are real constants. Using rank-one update I get ...
-4
votes
1answer
39 views

prove Determinants are equal [closed]

given matrix $A$, matrix $B$ is the matrix we get after adding row $i$ plus a linear combination of other rows to the row $i$, in matrix $A$. Prove that : $detA=detB$
-1
votes
1answer
67 views

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

matrix M $n\times n$ given, which its left-right diagonal contains the numbers from $1$ to $n$, and all the other numbers equal $n$. calculate the determinant of the matrix M.
7
votes
1answer
77 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. ...
0
votes
4answers
40 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
votes
3answers
84 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.
1
vote
0answers
46 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
votes
0answers
22 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
votes
0answers
23 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\\ ...
1
vote
1answer
34 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 ...
0
votes
1answer
24 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
votes
1answer
37 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
29 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
201 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
57 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
59 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
63 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
26 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
27 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
68 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 ...
-1
votes
3answers
83 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
130 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 ...
2
votes
0answers
48 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
18 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
vote
3answers
58 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
votes
0answers
12 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
114 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 $$ ...
10
votes
3answers
198 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
26 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 ...
1
vote
4answers
48 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
632 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
36 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
54 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
90 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
32 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
vote
1answer
61 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
46 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
26 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
105 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
30 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 & ...
4
votes
2answers
60 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 ...