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.

learn more… | top users | synonyms

-1
votes
0answers
8 views

Finding Linear transformations

Hello everyone! I stumbled upon some difficulties while trying to solve some questions assigned. I am familiar with the concepts but I'm having difficulty on figuring out on how to solve the ...
0
votes
1answer
37 views

Please help with this demonstration of determinants

inductively prove that: $$D[A_n,A_{n-1},....,A_2,A_1]=(-1)^{n(n-1)/2} D[A_1,A_2,...,A_{n-1},A_n]$$ where $A_n$ is a column of matrix $A\in K^{n\times n}$ and $D[A...]$. where $D$ is the determinant ...
-2
votes
1answer
16 views

Square matrices and find all real numbers

so I'm stuck on these two problems which I'm trying to solve but having difficulty with. Can anyone give me a helping hand? This is my first course in linear algebra so I might be slow.
0
votes
0answers
34 views

What is an intuitive way to think of the determinate?

Specifically, what is an intuitive way to think of a determinate in terms of systems of equations and/or vectors? I've read on here before that the determinate has something to do with volume or ...
1
vote
1answer
58 views

Why don't all metrics have trivial determinant?

What is wrong with this argument? Let $V$ be a vector space and $g$ an inner product. There exists an orthonormal basis for $V$. That is, in this basis $(g_{ij})=I$. But then given any other basis, ...
0
votes
1answer
53 views

Show a matrix is invertible [duplicate]

How to show that $$A=\begin{pmatrix}1233&2344&1324&3456\\ 2342&11233&1432&13256\\234132&32432&1234567&43254\\423412&42354&452356&13245\end{pmatrix}$$ ...
0
votes
1answer
32 views

Prove that $dy_i=0$ for each $i$ [on hold]

Let $xy_i= \sum_{j=1}^n a_{ij}y_j, \forall i$ with $1 \leq i \leq n$. Let $d$ be the determinant of the system of these equations ( $\sum_{j=1}^n (\delta_{ij}x-a_{ij})y_j=0, \forall i$ with $1 \leq i ...
2
votes
1answer
37 views

Find the Least Integer $k$ such that $B^k=I$

If $A$ and $B$ are two non Singular Matrices such that $B\ne I$, $A^6=I$ and $$AB^2=BA$$ Then what is the Least Integer $k$ such that $B^k=I$ My Try: Given $$AB^2=BA$$ which we can write as ...
1
vote
1answer
74 views

Showing that $\det(AB)=\det A \det B$ with the following identity.

Given the following formulation of the determinant with Levi-Civita permutation symbols, show that $\det(AB)=\det A \det B$. $$\det A = \sum\limits_{ij\cdots l}\epsilon_{ij\cdots l} ...
0
votes
1answer
18 views

“Hadamard's Maximum Determinant Problem” What is the maximum determinant value of 3x3 matrix whose entries consist of only 3 and 0.

I'm currently studying linear algebra. I faced one question that bothers me so hard. The question is about Hadamard's Maximum Determinant Problem. Since I cannot understand the concept of this, I ...
1
vote
0answers
15 views

Hankel determinant involving Fibonacci numbers

Let $F_n$ denote the nth Fibonacci number, with $F_1 = F_2 = 1$. Denote by M(n) the nxn Hankel matrix with $i,j $ entry $F_{i+j-1}^{n-1}$, where i and j range from 1 through n. Finally, let d(n) = ...
0
votes
0answers
45 views

When can $|AB-I|=|BA-I|$?

Prove or disprove that for ANY two matrices $A$ (of dimension $m$ by $n$) and $B$ (of dimension $n$ by $m$), $\det(AB-I)=\det(BA-I)$. The answer is easily false as I found a counter example. ...
2
votes
1answer
55 views

Which determinant could we find?

$A$ and $B$ are matrices and I found the determinants of $$A + B,\, A - B,\, AB,\, A^{-1},\, B^T.$$ If we know the determinants of $A$ and $B$ but don't remember the matrices $A$ and $B$, which of ...
0
votes
4answers
39 views

If two invertible matrices agree on a vector, does this imply their determinant agrees as well?

As stated, if we let $A, B \in M_n(\mathbb{R})$ be invertible and there is some $v\in R^n$ such that $$Av = Bv$$ does it follow that $\det(A) = \det(B)$? Additionally, does this hold if we let $A, B ...
0
votes
3answers
41 views

Square root of determinant equals determinant of square root?

Is it true that for a real-valued positive definite matrix $X$, $\sqrt{\det(X)} = \det(X^{1/2})$? I know that this is indeed true for the $2 \times 2$ case but I haven't been able to find the answer ...
0
votes
2answers
31 views

Prove that Det(A-E)=0 if and only if AC=C

We have some $n \times n$ matrix $A$ and $n \times 1$ vector C. Let $E$ be the identity matrix. $$Det(A-E)=0 \iff AC=C.$$ Me and a few friends have been trying to prove it, but none of us could. ...
1
vote
1answer
28 views

About a determinant identity.

If $A$ is any matrix and $B$ is a rank $2$ matrix of the same dimension then it follows that for any real $t$, $det(A -B) = [1-\partial_p + \frac{1}{2}\partial_p^2 ]det(A + pB) \vert _{p=0}$ I ...
3
votes
2answers
49 views

If $A,B$ are square matrices and $A^2=A,B^2=B,AB=BA$, then calculate $\det (A-B)$

If $A,B$ are square matrices and $A^2=A,B^2=B,AB=BA$, then calculate $\det (A-B)$. My solution: consider $(A-B)^3=A^3-3A^2B+3AB^2-B^3=A^3-B^3=A-B$, then $\det(A-B)=0\vee 1\vee -1$ The result of ...
2
votes
1answer
47 views

Compute a determinant [closed]

I want to compute this determinant: $$ \begin{vmatrix} \sin(2x)&\sin(3x)&\sin(4x)\\ \sin(3x)&\sin(4x)&\sin(5x)\\ \sin(4x)&\sin(5x)&\sin(6x) \end{vmatrix} $$
0
votes
2answers
34 views

Are determinants functions, numbers or matrices?

Let $M$ be a matrix such that \begin{equation} M = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \end{equation} As I understand it, \begin{equation} \det(M) = \begin{vmatrix} a & b \\ ...
2
votes
0answers
51 views

Is the determinant an analytic function?

I came accross a paper stating that the analytical property of determinants of complex matrices allows us to use some theorem for analytic functions. I am not able to confirm this since I am not sure ...
0
votes
1answer
22 views

Can we deduce that there are infinitely many integers $n$ such that $detA_{n}=0$?

Let $δ_{n},θ_{n},ω_{n}$ be three real sequences converging to $δ,θ,ω$ respectively. Define the following matrix $$A_{n} = \begin{bmatrix} δ_{n-1} & θ_{n-1} & ω_{n-1} \\ δ_{n} & θ_{n} ...
5
votes
1answer
137 views

Determinant evaluation for matrix with $-1, 2, -1$ below/on/above diagonal [duplicate]

What is the trick for evaluating the determinant of this matrix? $$\begin{bmatrix} 2 & -1 \\ -1 & 2 & -1 \\ & -1 & 2 & -1 \\ && -1 & 2 & -1 \\ &&& ...
0
votes
3answers
31 views

Effect of row operations on determinant for matrices in row form

I understand that adding a multiple of one row to another in a matrix has no effect on the determinant, which seems to contradict something I learned earlier: if I understand correctly, for a $n\times ...
2
votes
1answer
50 views

$4\times4$ determinant trick

This link uses a trick to find the determinant of a $3\times3$ matrix that goes like this: Put a copy of the matrix next to it, and now consider this as a $6\times3$ matrix. Find the sum of the ...
3
votes
3answers
93 views

Find $\det(A^{2}+A^{T})$ when eigenvalues are $1,2,3$

We have to find $\det(A^{2}+A^{T})$. It is given that eigenvalues of $A$ are $1,2,3$. My attempt: Since the question implicitly states that the answer would be same for all $A$ with eigenvalues ...
0
votes
1answer
62 views

A faster way to tell if a matrix is not non-singular.

If an n by n square matrix 'W' has four elements in 2 rows and the same four elements in two columns , in other words a 2 by 2 'sub-matrix' ; if this sub-matrix is singular and has no two rows that ...
1
vote
2answers
49 views

Simple lower bound for a determinant

Let $A$ in $\mathbb{Q}^{n \times n}$ such that $\det(A) > 0$? Is there a simple lower bound for $\det(A)$ in terms of the entries of $A$? Edit: Motivation: Let $M$ be an $m \times n$ matrix. I ...
0
votes
1answer
43 views

Let $\rho : G \rightarrow GL_n(\mathbb{C})$ be a representation show that $|\operatorname{tr} X| \leq \dim \rho$

Let $G$ be a finite group. Let $\rho : G \rightarrow GL_n(\mathbb{C})$ be a representation, pick $g \in G$ and write $X=\rho(g)$. Prove that all eigenvalues of $X$ are roots of unity, and deduce that ...
0
votes
1answer
41 views

LU Decomposition - Are there multiple ways to calculate?

I am attempting to use LU Decomposition to calculate the determinant of a matrix. Given: $$ A = \begin{pmatrix} 1 & 2 \\ 5 & 6 \end{pmatrix} $$ When using this calculator: Here the values ...
1
vote
1answer
31 views

Lower bound for the size of a determinant

Given a matrix $A$ in $\mathbb{R}^{n \times n}$ and let $a_{\min} = \min_{\substack{1 \leq i \leq n \\ 1 \leq j \leq n}} \{ |A_{ij}| \}$. Does $$ a_{\min} \leq \det(A) $$ always hold?
0
votes
1answer
35 views

Determinant is product of different primes

Let $M_{1}$, $M_{2}$ be two $n \times n$ matrices with entries in $\mathbb{Z}$ such that $\det(M_{1})=\det(M_{2}) = p_{1}p_{2}\cdots p_{m}$, where $p_{j}$ are distinct prime numbers. I need to show ...
0
votes
0answers
24 views

Computing a lower bound for the minimal componentwise distance of vertices of polyhedra

Let $A$ be a matrix in $\mathbb{R}^{m \times n}$ and let $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$ be a polytope. I want to compute a lower bound on the minimal componentwise distance of two ...
1
vote
2answers
91 views

Prove that $\det(A^p+B^p)=\det (A^p)+\det(B^p) +\operatorname{tr}\left(\left(A\operatorname{adj}(B)\right)^p\right)$

Let $A,B$ be $2\times 2$ matrices such that $AB=BA$. Prove that for every positive integer $p$: $$ \det(A^p+B^p)=\det (A^p)+\det(B^p) ...
0
votes
1answer
39 views

Does there exist a simple solution to the following eigenvalue problem

I am looking for the values of $Z$ for which the determinant of the following $N$-dimensional matrix vanishes: \begin{equation} \begin{bmatrix} N(1-Z) & N-1 & N-2 & \cdots & \cdots ...
0
votes
0answers
23 views

What is the fastest method for finding the determinant of any square matrix?

There are several methods to find the determinant of a matrix. What is the fastest method to fastest for finding the determinant of any square matrix. Any square matrix being a matrix that is ...
1
vote
0answers
26 views

Hypermatrices, hyperdeterminants and Grassmannians.

Let $Gr(k,n)$ the Grassmannian manifold of the $k$-planes in $\mathbb{C}^n$ and consider the Plucker embedding $\pi: Gr(k,n) \to \mathbb{P}(\Lambda^k \mathbb{C}^n)$. Let $A$ be the set of $n \times n$ ...
1
vote
4answers
30 views

Find matrix from Eigenvectors and Eigenvalues

A matrix $A$ has eigenvectors $v_1 = \left( \begin{array}{c} 2 \\ 1 \\ \end{array} \right)$ $v_2 = \left( \begin{array}{c} 1 \\ -1 \\ \end{array} \right)$ with corresponding ...
1
vote
0answers
22 views

Powers of coefficients divide the resultant

Let $f(x)=a_0x^n+a_1x^{n-1}+\dots+a_n$, $g(x)=b_0x^m+b_1x^{m-1}+\dots+b_m$, with coefficients in a field. Prove that $a_0^mb_m^n$ divides the resultant of $f(x)$ and $g(x)$. I have written the ...
3
votes
4answers
372 views

When a determinant is zero

Is it true that if $C$ is a square matrix of size $n$ and $\det(C) = 0,$ then $C^n = O_n$ or the $0$ matrix? If yes, then why is that? I know that the reverse is obviously true, so I wondered if ...
0
votes
1answer
46 views

Calculating the Jacobian of inverse functions

The task is this: given the following pair of functions: \begin{cases} u = e^x cos(y) \\ v = e^x sin(y) \end{cases} Determine the inverse functions, and compute the Jacobian of the inverse functions ...
2
votes
0answers
34 views

Is the Cone over Grassmannian manifold a determinantal variety?

Let consider the Grassmann manifold $Gr(k,n)$ in the Plucker embedding and the Cone over $Gr(k,n)$, say $C(Gr(k,n))$. On the other hand consider $M$ the set of $n \times n$ skew-symmetric matrices. ...
1
vote
1answer
51 views

Determinant of the symmetric part of a matrix.

Define the symmetric part of a matrix $A$ as: $$ A^+ := \frac{A+A^t}{2}. $$ Is there a formula relating the determinants of $A$ and $A^+$? Thanks!
1
vote
0answers
55 views

Invertability of a matrix

$\newcommand{\AA}{\mathbf{A}} \newcommand{\Tr}[1]{\operatorname{Tr}\left[#1\right]}$ I have a problem that I suspect there is a “relatively” simple answer to but it is currently eluding me. I am ...
6
votes
2answers
351 views

How can I quickly find the determinant of this matrix

$$ \begin{vmatrix} 14 & 2 & 1 & 3\\ 31 & 4 & 5 & 6\\ 26 & 3 & 7 & 4\\ 10 & 1 & 3 & 2\\ \end{vmatrix} ...
0
votes
1answer
14 views

Characteristic Polymonmial 4x4 Matrix

I have to find the characteristic polynomial to find Jordan normal form. I chose to solve this via column expansion on the first determinant, and then row expansion in the inner determinant. But ...
0
votes
1answer
32 views

LU Decomposition

I'm having trouble understanding which answer is correct. I'm currently reading a paper: lecture 12 - They give the following example: Let: $$ A = \begin{bmatrix} 1&2&3 \\ 2&5&12 ...
5
votes
4answers
292 views

Decompose this matrix as a sum of unit and nilpotent matrix.

Show that the matrix $A=\begin{bmatrix} 1 & 0 \\ 2 & 1 \\ \end{bmatrix}$ can be decomposed as a sum of a unit and nilpotent matrix. Hence evaluate the matrix ...
0
votes
1answer
27 views

is the Jacobian Determinant continuous

Is the Determinant of the Jacobian a continuous function? i.e. $$f:\mathbb{R}^n \rightarrow \mathbb{R}^n $$ $$ \forall \varepsilon >0 \quad \exists \delta >0 : |x-x_0 |<\delta ...
0
votes
0answers
12 views

Question on a proof concerning a Sylvester matrix and the roots of polynomials

For clarity here, $R(f,g)$ is the determinant of a Sylvester matrix. Also, the author writes a polynomial as $f(x) = a_{0}x^n + ... + a_{n}$. Whats the reason as to why $\lambda = 1$? I don't ...