Determinant Definition and 494 Threads

Hi Folks, I have a question about determinants that is probably quite simple. I know that if you have a matrix and you interchange rows, the determinant changes. However, if you cyclically change the rows up or down, still in order, the determinant does not change. What is the theorem or...

Hello all. I have a question about determinants. The question is from an exam and solutions were not published. I would like to know if my solution is correct. Please excuse me for the imperfect formatting, I am struggling with the interface. Espeacially the superscripts were supposed to be...

Homework Statement Given that A is any 2x2 matrix show that it is invertible if and only if det(A) \neq 0. Homework Equations The Attempt at a Solution If A is invertible then we know there exists an inverse matrix, say B, such that AB = BA = I. It follows that...

Hi all, I'm trying to get to grips with the Frechet derivative, and whilst I think I've got all the fundamental concepts down, I'm having trouble evaluating some of the trickier limits I've come up against. The two I'm struggling with currently are the further derivatives of the functions...

Homework Statement Hello all, I have a parameter x of a matrix M = [21, 7 14, 7] the determinant of |(M - xI)| = 0 Homework Equations I is known as the identity matrix The Attempt at a Solution My solution seems to be a bit...

Hello, I am quite new to Geometric Algebra, this is the reason for the silly question. In Geometric Algebra, the following implicit definition of determinant is given: f(\mathbf{I_n})=det(f)\mathbf{I_n} where f is a linear function extended as an outermorphism, and \mathbf{I_n} is the unit...

Hi, I'm trying to see why the following theorem is true. It concerns the derivative of the log of the determinant of a symmetric matrix. Here's the theorem as stated: For a symmetric matrix A: \frac{d}{dx} ln |A| = Tr[A^{-1} \frac{dA}{dx}] Here's what I have so far, I'm almost at the...

Use determinant property to evaluate matrice determinant Homework Statement 3x3: A = 12 27 12 28 18 24 70 15 40 What determinant property can I use to find the determinate of this matrix? I cannot see any relationship between the rows or columns. Apparently 720 comes into it somehow, but I...

The question is: Compute the determinant of the n × n matrix A for which aij = max(i, j). I can compute determinants but I don't really know what the last bit means. Any help appreciated.

If I have a matrix A, and I use n different row operations of this form: a_kR_i + R_j \rightarrow R_i to construct a new matrix B, what is the determinant of A in terms of B? Solved! |A|=|B|\prod^n_{k=1}\frac{1}{a_k}

Homework Statement Given n2 functions fij, each differentiable on an interval (a,b), define F(x) = det[fij] for each x in (a,b). Prove that the derivative F'(x) is the sum of the n determinants, F'(x) = \sum_{i=0}^n det(Ai(x))$. where Ai(x) is the matrix obtained by differentiating the...

Let's say I have a matrix A: A=\begin{bmatrix} f(x)& z_1(x)& z_2(x)\\ 0& a(x)& b(x)\\ 0& c(x)& d(x) \end{bmatrix} I've noticed that the determinant of A will either be a(x)d(x) - b(x)c(x) or f(x)a(x)d(x)-f(x)b(x)c(x). I've never found an example of it taking another form. My question is, is...

Why is the determinant of a mixed state density matrix always positive? In the specific case of a 2-dimensional Hilbert space, the density matrix (as well as any other hermitian matrix) can be expressed as \rho=\frac 1 2 (I+\vec r\cdot\vec \sigma) so its determinant is...

Homework Statement so my problem is to calculate the determinant of this matrix \left[\begin{array}{ccccc} 1 & 2 & 3 & 3 & 5 \\ 3 & 2 & 1 & 2 & 2 \\ 1 & 2 & 3 & 4 & 5 \\ -1 & 0 & -8 & 1 & 2 \\ 7 & 2 & 1 & 3 & 2 \end{array}\right] Homework Equations The Attempt at a...

Let A, B and C be 3x3 invertible matrices where det(A)=4 , det(B)=4 and det(C) is some non-zero scalar. a) det [(C^T)(A^-1)(B^2)(C^-1)] b) det [-2(A^2)^T(C^2)(B^-1)(C^-1)^2] a) What I got is: det [(A^-1)(B^2)(C^T)(C^-1)] = det [(A^-1)(B^2)(C)(C^-1)] = det [(A^-1)(B^2)] =...

Just wondering about the "structure" of a determinant... How much can a determinant tell you about the entries of a matrix? How much more if you know the size of the aforementioned matrix? How much more if you know that the matrix is symmetric?(perhaps a silly question). How much more if you...

There is a surface defined by setting implicit function g(x)=0, where x is a 3 by 1 column vector, denoting a point on the surface; 3X1 vector \nablag(x) is the Gradient(surface normal at point x; 3X3 matrix H(g(x)) = \nabla^2(g(x)) is the Hessian Matrix; 3X3X3 tensor...

Find the determinant of C by first row reducing it to a matrix with first column 1,0,0,0. Show the row operations and explain how all this tells you the value of the determinant of C when you are done. C=(2,0,-6,8;3,1,0,3;-5,1,7,-8;0,0,5,1) where ; indicates a new row. We're suppose to...

Let ∑ be the variance-covariance matrix of a random vector X. The first component of X is X1, and the second component of X is X2. Then det(∑)=0 <=> the inverse of ∑ does not exist <=> there exists c≠0 such that a.s. d=(c1)(X1)+(c2)(X2) (i.e. (c1)(X1)+(c2)(X2) is equal to some...

Homework Statement Let A, B and C be 3 x 3 invertible matrices where det A = -2 ,det B = -2 and det C is some non-zero scalar. Then det (CTA−1B2C−1) = ? and det [ −2(A2)TC2B−1(C−1)2] = ? the T represents transpose and the -1 represents inverse. Homework Equations What does...

I'm having trouble solving this question: If det | a b c | | p q r | = 2, | u v w | find det (3B^-1) where B = | 2u 3u-a a-p | | 2v 3v-b b-q | | 2w 3w-c c-r | Is it possible to transpose B? So far what I got is 4; however, the instructor told me that...

Hi all! I was working on some homework for the linear algebra section of my "Math Methods for Physicists" class and was studying skew symmetric matrices. There was a proof I saw on Wikipedia that proves that the determinant of a skew symmetric matrix is zero if the number of rows is an odd...

how can i show that a determinant is divisible by a number, without directly evaluating the determinant?

How is it the determinant of an orthogonal matrix is \pm1. Is it: Suppose Q is an orthogonal matrix \Rightarrow 1 = det(I) = det(QTQ) = det(QT)det(Q) = ((det(Q))2 and if so, what is it for -1. Thanks.

How can I get the determinant of this matrix? 1-n 1 ...1 1 1 1-n ...1 1 . . . . . . . . 1 1 ... 1 1-n I think that the answer is 0 but... why? Thank you.

HOw can this determinant be 0 ?? let be a 3x3 matrix with elements a_{1,j}= j a_{2,j}= j+3 a_{3,j}= 6+j with i=1,2,3 and j=1,2,3 ... but DetA=0 apparently there is no zeros or other condition that points that determinant should be 0, is there any explanation ??

Homework Statement This is all in summation notation. Given a 3x3 matrix A_{ij}, show that det[A]=1/6(A_{ii}A_{jj}A_{kk}+2A_{ij}A_{jk}A_{ki}-3A_{ij}A_{ji}A_{kk}) Homework Equations I've been told that we're supposed to begin with det[A]=1/6\epsilon_{ijk}\epsilon_{pqr}A_{ip}A_{jq}A_{kr}...

Hi, I'm having some problems with the derivation of the Jacobian determinant when used to describe co-ordinate transformations. As I understand it, the Jacobian determinant should relate the areas defined by two vectors in both co-ordinate systems. As the vectors are not necessarily...

Homework Statement Let R be the field of real numbers, and let D be a function on 2x2 matrices over R, with values in R, such that D(AB) = D(A)D(B) for all A, B. Suppose that D(I) != D ([0 1 1 0]) Prove that a) D(0) = 0 b) D(A) = 0 if A2= 0 c) D(B) = -D(A) if B is obtained by interchanging...

I just started reading the first chapter of Georgi Shilov's "Linear Algebra" and I have a question about his notation for determinants. His notation, (7), for the determinant of an n x n matrix seems to be \det ||a_{ij}||. (4) suggests Shilov would write the 1 x 1 matrix with the single...

Homework Statement Show without evaluating the determinant the equality. Homework Equations \left( \begin{array}{ccc} 1 & a & bc \\ 1 & b & ac \\ 1 & c & ab \end{array} \right) = \left( \begin{array}{ccc}...

I've been going through properties of determinants of matrices and found the following: Assuming products are defined and the matrices involved are nonsingular of the same order The determinant of the product of any number of matrices is equal to the determinant of each matrix; where the order...

Hi, We use as an integration form in Riemannian geometry the covariant \int \sqrt{g}d\Omega I understand how this is invariant under an arbitrary change of coordinates (both Jacobian and metric square root transformation coefficient will cancel each other), what I don't understand is why don't...

Homework Statement If A and C are nxn matricies, with C invertible, prove that det(A)=det((C^-1)AC). Homework Equations The Attempt at a Solution I think the way to go is to show that if A=(C^-1)AC, then det(A)=det((C^-1)AC), but I'm not sure how to show A=(C^-1)AC. I know...

Homework Statement I don't understand why the determinant of a matrix is equal to its transpose...how is this possible? Homework Equations The Attempt at a Solution

Homework Statement It's my understanding that 2X2 and 3X3 determinats kinda measure volume...is there a general interpretaion for an nxn determinant ( in words, not formulas please) Homework Equations The Attempt at a Solution

Hello, I don't want how to prove: Matrix A is nilpotent, so A^k=0. Prove that det(A+I)=0. Thank you so much :-)

Homework Statement Find the determinant of the following matrix 0 1 1 ... 1 1 0 1 ... 1 1 1 0 ... 1 . . . 1 1 1 ... 0 The zeros are always at the diagonal. All other points are 1. The Attempt at a Solution Consider the matrix as block matrices. We know that the determinant of...

definition of the determinant i = 1" Lemma:Let B be an element in M_n_x_n(F), where n >= 2. If row i of B equals e_k for some k (1<= k <= n ), then det(B) = (-1)^i^+^kdet(B_i_k). Proof: The proof is by mathematical induction on n. The lemma is easily proved for n = 2. Assume that for some...

I thought this problem was pretty straightforward, but I can't seem to match the answers in the back of the book. The problem is: Find the determinant of the following linear transformation. T(v) = <1, 2, 3> x v (where the x means cross product) from the plane V given by x + 2y +...

Homework Statement find the determinant using row operations: 1 -2 2 0 5 -1 2 -4 1 Homework Equations The Attempt at a Solution i took row 3 and took 2 x row 1 away from it to get : 1 -2 2 0 5 -1 0 0 -3 1 x 5 x (-3) = -15...but i multiplied a row by 2 so i should get -30...

Homework Statement If A is an idempotent matrix (A^2 = A), find all possible values of det(A). Homework Equations The Attempt at a Solution I'm not sure if this is the proper way to show it, but here's what I did: Since A = A^2, det(A)=det(A^2) So det(A) = det(A)*det(A)...

Homework Statement Prove that \Gamma^\mu_{\mu\lambda}=\frac{1}{\sqrt{-g}}\partial_\lambda(\sqrt{-g}) where g is the determinant of the metric, and \Gamma are the Christoffel connection coefficients. The Attempt at a Solution From the general definition of the coefficients I got...

Homework Statement Find the eigenvalues of B = [5 2 0 2], [3 2 1 0], [3 1 -2 4], [2 4 -1 2]. Compute the sum and product of eigenvalues and compare it with the trace and determinant of the matrix. Homework Equations The Attempt at a Solution I get the characteristic polynomial...

Hi, I'm a reading a thesis on Generalised Complex geometry and it mentions an object " det V" and "det V*", for a real vector space V, and its dual V*. Could anyone tell me what this notation means? I've been unable to find anything mentioning a determinant of an entire vector space, so I'm...

Homework Statement Hi, i'm trying to solve this problem: http://img4.imageshack.us/img4/3876/53065718.jpg .[/URL] The Attempt at a Solution I have shown it for n=2 and n=3 then I was going to use induction to prove it for all n, but I can't seem to find a way to do it. Please...

how to calculate determinant of \frac{\partial}{\partial t} \delta (t-\tau)?

can anyone explain or prove this?? Ax={(xTAT)T} how can a determinant of a matrix become an area?? example: 2 X 2 matrix the determinant of this matrix is ad-bc ! but i search on wikipedia it wrote like this :The assumption here is that a linear transformation is applied to row...

Can someone please thoroughly explain how the determinant comes from the wedge product? I'm only in Cal 3 and Linear at the moment. I'm somewhat trying to learn more about the Wedge Product in Exterior Algebra to understand the determinant on a more fundamental basis. A thorough website or...

Is there a definition of determinant of a non - square matrix??