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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4
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0answers
46 views

Rank of a matrix whose all entries have the form $m^k$

The original problem is: Compute the determinant $$\begin{vmatrix} 1^k & 2^k & 3^k & \cdots & n^k \\ 2^k& 3^k & 4^k &\cdots & (n+1)^k \\ 3^k& 4^k ...
-2
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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
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0answers
44 views

What is an intuitive way to think of the determinate? [duplicate]

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
74 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
66 views

Show a matrix is invertible [duplicate]

How to show that $$A=\begin{pmatrix}1233&2344&1324&3456\\ 2342&11233&1432&13256\\234132&32432&1234567&43254\\423412&42354&452356&13245\end{pmatrix}$$ ...
2
votes
1answer
44 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
85 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
40 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
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0answers
22 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
52 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
58 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
41 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
48 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
34 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
30 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
59 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
51 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
36 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
56 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
24 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
162 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 \\ &&& ...
1
vote
2answers
42 views

Effect of row operations on determinant for matrices in row form [duplicate]

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
58 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
109 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
88 views

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

If an n by n square matrix 'W' has an r by r sub-matrix that is singular and (n-1) > r > n/2 when is it true the whole matrix is also singular? Maybe this could in some cases show a matrix is ...
1
vote
2answers
56 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
46 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
57 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
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1answer
34 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?
1
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1answer
40 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
28 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
111 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
41 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
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0answers
28 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
43 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
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4answers
63 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
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0answers
23 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
387 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
60 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
40 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
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1answer
56 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
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0answers
66 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
387 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
33 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
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1answer
40 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
335 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
44 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
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0answers
22 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 ...
1
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2answers
39 views

Inequality involving Hadamard's inequality

Let $A$ be matrix in $\mathbb{R^{m \times n}}$. Let $A$ and $B$ be quadratic submatrices of $M$ such that $\det(A)< \det(B)$. Does this imply $\prod_{i=1}^n \|A^i\| < \prod_{i=1}^n \|B^i\|$ ...
6
votes
3answers
52 views

Calculation of determinant

Is there any easier way to make sure the determinant of the following matrix is n (the dimension of square matrix)? $ \begin{vmatrix} 1 & -1 & -1 & -1 & \cdots & -1 \\ 1 ...