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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1answer
37 views

Do we have $\det=e^1\wedge\cdots\wedge e^n$?

If we think of the determinant as a multilinear map from the set of $n$-column vectors to $\mathbb{R}$, $$\det:\mathbb{R}^n\times\cdots\times\mathbb{R}^n\to\mathbb{R},$$ am I right in saying that ...
0
votes
0answers
23 views

Can the Berezinian be used instead of the determinant of a block matrix?

Suppose we have a $2N \times 2N$ matrix $(N\ge 2)$. For example we can consider a block matrix: $$X= \left[\begin{matrix} A & B \\ C & D \end{matrix}\right]$$ with $A,B,C,D=$ $2\times 2$ ...
4
votes
1answer
76 views

Series Expansion of the determinant for a matrix near the identity.

The problem is to find the second order term in the series expansion of the expression $\mathrm{det}( I + \epsilon A)$ as a power series in $\epsilon$ for a diagonalizable matrix $A$. Formally we ...
2
votes
1answer
16 views

Is the function that maps a matrix to the determinant of a submatrix continuous?

Let $M$ be the space of $m \times n$ matrices over $\mathbb{R}$. For each $A$ in $M$ let $A'$ be a fixed submatrix of $A$. Is the function $M \to \mathbb{R}$ defined by $A \mapsto \det(A')$ ...
0
votes
0answers
12 views

Recursive equations with matrices, and a question about determinants in relation to power of matrices

If we have the matrix equation $AX^{(i)} = X^{(i+1)}$ where $A$ is a constant matrix, this is what we'd call a recursive function; in matrix form. Moreover, if $X^{(i+1)} = X^{(i)}$, i.e. $AX = X$ ...
0
votes
1answer
80 views

Show that a 2x2 matrix A is symmetric positive definite if and only if A is symmetric, trace(A) > 0 and det(A) > 0

I need to show two parts of the implication are true. First: if $A$ is $2\times 2$ and is symmetric positive definite then $trace(A)>0$ and $det(A)>0$. Second: if $trace(A)>0$ and ...
-1
votes
1answer
27 views

Show that any orthogonal matrix has determinant 1 or -1 [duplicate]

Hello fellow users of this forum: Show that for any orthogonal matrix Q, either det(Q)=1 or -1. Thanks
1
vote
1answer
37 views

Determinant of a matrix with 2x2 blocks

I have a matrix, say $A$ and want to find it's determinant $detA$. A is $L\times L$ and made up of $2\times 2$ blocks $M_{i,j}$ giving it a total size of $2L \times 2L$. The entries of the blocks ...
0
votes
0answers
14 views

prove that $X$ is invertible if and only if $Y$ is invertible. if $(-1)^i(1+i)x_i^T=y_i$

$X=[x_1,x_2,...,x_n]$ and $Y =$ $y_1\\y_2\\...\\...\\...\\y_n$ where $x_i$ and $y_i$ are column and row matrices respectively. $X$ and $Y$ are both $n$ x $n$ matrices. if $$(-1)^i(1+i)x_i^T=y_i$$ ...
0
votes
0answers
10 views

Identity with discriminant

Let $K$ be a field which is finite or of characteristic zero, let $L$ be an extension of finite degree $n$ of $K$, and let $σ_1,…σ_n$ be the $n$ distinct $K$-isomorphisms of $L$ into an algebraically ...
0
votes
0answers
48 views

Can one define wedge products using determinants for $n$-forms?

I was talking to Ted Shifrin in math chat yesterday and he mentioned there is a way to define wedge products using determinants. As far as I understand, given a set of vectors $x,y,z,v,u... \in ...
0
votes
0answers
12 views

General Discriminant Formula for Multivariate Polynomials over Reals.

Consider \begin{eqnarray} b'(I_{s}c-A)b>0 \end{eqnarray} where $A$ is a symmetric, positive definite s by s square and I is the identity and c is a constant. Solutions are to be found using $b ...
7
votes
0answers
180 views

Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix. I have already asked a (viewed but unanswered) ...
4
votes
0answers
45 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
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
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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
73 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
63 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
42 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
82 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
33 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
20 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
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0answers
51 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
57 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
43 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
29 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
55 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
49 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
55 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
152 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
2answers
34 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
55 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
101 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
78 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 then the whole matrix is singular. Is this never the case? Or are there many instances where this is true? ...
1
vote
2answers
51 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
52 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
38 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
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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
98 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
24 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
39 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
45 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
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 ...