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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71 views

determinant of the symmetric matrix $8\times8$

How to compute the determinant of the following matrix: $ \left( \begin{array}{cccccccc} 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 1 & 1 & 1 & 1 & ...
2
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1answer
135 views

Looking for a proof that the resultant is the product of the differences of roots

I'm trying to find a general proof to an exercise given in Garrity et al's book, Algebraic Geometry: A problem-solving approach. The problem is this: Given two polynomials f and g, show that for ...
3
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1answer
80 views

A determinant coming out from the computation of a volume form

I am convinced that the following identity is true: \begin{equation} \det\begin{bmatrix} 1+a_1^2 & a_1 a_2 & a_1 a_3 & \ldots & a_1a_n \\ a_1a_2 & 1+a_2^2 & a_2a_3 & ...
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0answers
47 views

Determinants, Pfaffians, and…?

I recently stumbled across the wikipedia entry on Pfaffians and found them rather interesting, especially the property below. (assuming $A$ is a $2n\times 2n$ skew symmetric matrix) ...
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1answer
139 views

Eigenvalue of the sum of a symmetric matrix and the outer product of it's eigenvector

I have a symmetric matrix $A$ with eigenpairs $(\lambda_k, v_k)$ with $k \in (1,..,n)$. A new matrix $B$ is made from an eigenpair $(\lambda_i, v_i)$ like this: $$B = A - \lambda_i v_i v_i^T$$ where ...
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1answer
334 views

Change in determinant when multiplying row of a matrix

I'm a bit confused with something I read and I hope you can help me. I'm studying determinants and right now how matrix row operations change the determinants. I read (and in fact quote): the effect ...
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1answer
58 views

Solving a system of equations

(Image Attached) I've begun with the hint and found out that $(det\ A)*x=adj\ A*c$ and therefore what x is. My question would be how would I go about finding what $det\ A_i$ is? Should I go about ...
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0answers
46 views

Linear program of 0-1 knapsack problem and proof of integer

I have some questions about the knapsack problem. How can the 0-1 knapsack problem described as a linear program? How to proof that the solution of the 0-1 knapsack problem are integer? (I'm ...
5
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0answers
125 views

calculation of the determinant of a block matrix little help

I need to prove $$\operatorname{det}\begin{pmatrix}A & B \\ C & D\\ \end{pmatrix}= \operatorname{det}(DA-CB),$$ where $A,B,C,D \in M_{n\times n}(R)$ with the property that $A$ and $B$ ...
8
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1answer
756 views

Upper Triangular Block Matrix Determinant by induction

We want to prove that: $$\det\begin{pmatrix}A & C \\ 0 & B\\ \end{pmatrix}= \det(A)\operatorname{det}(B),$$ where $A \in M_{m\times m}(R)$, $C \in M_{m\times n}(R)$,$B \in M_{n\times n}(R)$ ...
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1answer
264 views

Extending a Chebyshev-polynomial determinant identity

The following $n\times n$ determinant identity appears as eq. 19 on Mathworld's entry for the Chebyshev polynomials of the second kind: $$U_n(x)=\det{A_n(x)}\equiv \begin{vmatrix}2 x& 1 & 0 ...
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1answer
84 views

Find the computational cost associated with calculating determinant of an $n\times n$?

How to determine the computational cost associated with calculating determinant of an $n\times n$ matrix, using LU factorization.
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2answers
328 views

Find the matrix given the determinant

Is there a general method to find a 3x3, or 2x2 matrices, given the determinant? I want to do a project with my students when we start to study Systems of Equations. It would be interesting if the ...
2
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1answer
65 views

Determinant using Leibniz formula

$$\begin{matrix} * & * & *&*&* \\ *&*&*&*&*\\ 0&0&0&*&* \\ 0&0&0&*&* \\ ...
2
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1answer
303 views

Cholesky, Inverse, and Determinant when updating the diagonal of a symmetric positive definite matrix

Suppose that $A$ is a symmetric positive definite matrix and assume its dimension $n$ is large. Let $I$ be the $n \times n$ identity matrix and $m \neq 0$ be a scalar. I'm interested in computing as ...
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1answer
58 views

$d+1$ distinct points of a rational normal curve in $\mathbb{P}^{d}$ are linearly independent

Let $X\subset\mathbb{P}^{d}$ be a rational normal curve. After a change of coordinates, it is the image of the map: $\nu:\mathbb{P}^{1}\rightarrow\mathbb{P}^{d}, (a_{0}:a_{1})\mapsto ...
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2answers
49 views

Finding a Matrix from Determinants

I've stumbled upon this problem on my homework, and I have no clue how to do it, and haven't found any help online: If I'm understanding this correctly, then $det(M) = ad - cb + eh - gf$ ? What I ...
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2answers
71 views

matrices vector spaces

Consider the vector space of 3 by 3 matrices with real coefficients. Let W denote the subset of matrices with determinant 0. Decide whether W is a subspace or not.
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2answers
67 views

Why adding a row with another row in square matrix A doesn't change the $\det(A)$ value?

Why adding a row with another multiplied row in square matrix $A$ doesn't change the $\det(A)$ value?
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1answer
63 views

Need help with a determinant problem [closed]

I'm learning determinants and just came across a problem. I've been trying really hard to solve it but no success so far. I just know that the answer is (3) 1 but don't know how to solve it? Please ...
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0answers
38 views

Matrices over $\Bbb{R}$ with $2\times 2$ skew-symmetric blocks

For any complex number $z\in{\Bbb C}$, define a $2\times 2$ matrix $\hat z$ as $$ \hat z:=\begin{pmatrix} a&-b\\ b&a \end{pmatrix} $$ where $z=a+ib$, $a,b\in{\Bbb R}$. Let $A=(z_{ij})$ be an ...
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2answers
82 views

$LDL^T$ decompositon of a symmetric matrix and a matrix determinant expression for the lower triangular entries

Let $n$ be a positive integer, and let $M$ be an integral, symmetric, nonsingular matrix. As $M$ is nonsingular, there exists an $LDL^T$ decomposition such that $D = (d_j)$ is diagonal and ...
4
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1answer
44 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 ...
4
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1answer
306 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
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1answer
26 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 $A \mapsto \det(A')$ ...
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0answers
19 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. $AX = X$ ...
2
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1answer
3k 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 ...
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1answer
49 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
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1answer
52 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 ...
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0answers
129 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 ...
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0answers
291 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
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0answers
49 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 ...
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0answers
48 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 ...
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1answer
78 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, ...
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1answer
88 views

Show a matrix is invertible [duplicate]

How to show that $$A=\begin{pmatrix}1233&2344&1324&3456\\ 2342&11233&1432&13256\\234132&32432&1234567&43254\\423412&42354&452356&13245\end{pmatrix}$$ ...
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1answer
50 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 ...
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1answer
168 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} ...
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1answer
128 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 ...
3
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1answer
43 views

Hankel determinant involving Fibonacci numbers

Let $F_n$ denote the $n$-th Fibonacci number, with $F_1 = F_2 = 1$. Denote by $M\left(n\right)$ the $n \times n$ Hankel matrix with $\left(i,j\right)$-th entry $F_{i+j-1}^{n-1}$, where $i$ and $j$ ...
2
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1answer
61 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 ...
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4answers
45 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 ...
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3answers
93 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 ...
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2answers
46 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. ...
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1answer
31 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
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2answers
88 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 ...
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1answer
67 views

Compute a determinant [closed]

I want to compute this determinant: $$ \begin{vmatrix} \sin(2x)&\sin(3x)&\sin(4x)\\ \sin(3x)&\sin(4x)&\sin(5x)\\ \sin(4x)&\sin(5x)&\sin(6x) \end{vmatrix} $$
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2answers
39 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
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0answers
82 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
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1answer
26 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
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1answer
236 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 \\ &&& ...