Determinant Definition and 494 Threads

Homework Statement A square (nn) matrix is called skew-symmetric (or antisymmetric) if AT = -A. Prove that if A is skew-symmetric and n is odd, then detA = 0. Is this true for even n? Homework Equations Det(A) = Det(AT) where AT= the transpose of matrix A The Attempt at a...

Homework Statement http://puu.sh/1rcsO I got the first one from a simple scaling, but I can not figure out the second one. Homework Equations Det(cA) = cDet(A) Scaling a row scales the determinant Adding rows/columns to each other does not affect the determinant Det(AT) = Det A...

Homework Statement Exercise 44 - In the picture attachedHomework Equations HINT - expand the expression for n and plug the result into equation (70), then use equation (63) n=(C-B)X(B-A) n dot (r - A) = 0 (eq. 70) A dot (B X C) = det {A B C} (eq. 63)...

Determinant -- best way of introducing determinants on a linear algebra course What is the best way of introducing determinants on a linear algebra course? I want to give real life examples of where the determinant is applied.

Hello, Typically the area of a parallelogram if give by A=b*h The det(M) =ad-bc, where m=2x2 matrix. How they are related? -- Shounak

Homework Statement Define L: R(mxm) to R(nxn). If L(A)=L(B), prove or disprove that det(A)=det(B). Homework Equations The Attempt at a Solution I think I can prove that this is true. L(A)=L(B) means that L(A)-L(B)=L(A-B)=0. Now let C be the matrix representation of L. We...

1(a) Construct a 4*4 matrix whose determinant is easy to compute using cofactor expansion but hard to evaluate using elementary row operations. (b) Construct a 4*4 matrix whose determinant is easy to compute using elementary row operations but hard to evaluate using cofactor expansion

Sorry for the format, I'm on my phone. Lets say the matrix is | 1 1 1 | | a b c | | a^2 b^2 c^| Or {{1,1,1} , {a, b,c} , {a^2, b^2,c^2}} Show that it equals to (b-c)(c-a)(a-b) I did the determinant and my answer was (bc^2) - (ba^2) - (ac^2) + (ab^2) + (ca^2) - (cb^2)...

Need a lot of help here guys. I need to prove that for an nxn matrix A, if i interchange two rows to obtain B, then det=-detA I have proved my basis (below), but I'm stuck on the hard part, the induction (which I'm required to do). I understand the steps of induction, but i don't know how...

Homework Statement Find the determinant of the matrix {{cos 25°, sin° 65}, {sin 120°, cos 390°}} (sorry, can't latex). {cos 25°, sin° 65} is first row and {sin 120°, cos 390°} is the second one. Homework Equations cos(a + b) = (cos a)(cos b) - (sin a) (sin b) The Attempt at a...

Hi all, This is a beginning step in proving aXb=|a||b|sin(theta) thank you

Number of rows>number of columns. Just out of curiosity, I've never seen this done before. I don't even know how if it were possible. Same with an mxn matrix where n>m+1. I don't think you would be able to find the determinant of this either.

Hello guys, can someone help me with this question please?

I'm trying to find a way to prove that the determinant of the transpose of an endomorphism is the determinant of the original linear map (i.e. det(A) = det(Aᵀ) in matrix language) using Dieudonne's definition of the determinant expressed in terms of an alternating bilinear form but am having...

Hello, So, given two points, x and x', in a Lorentzian manifold (although I think it's the same for a Riemannian one). If in x the determinant of the metric is g and in the point x' is g'. How are g and g' related?By any means can g=g'? In what conditions? I'm sorry if this is a dumb...

There's a geometric interpretation of the determinant of an operator in a real vector space that I've always found intuitive. Suppose we have a n-dimensional real-valued vector space. We can plot n vectors in an n-dimensional Cartesian coordinate system, and in general we'll have an...

Okay so I'm a first year engineering student and I'm taking linear algebra. I understand how to take determinants of nxn matrices, and I know how to do co-factor expansion. But I still don't understand what a determinant is. I don't like learning algorithms on how to do certain things in...

Homework Statement How can I prove that given two nXn positive semi-definite matrices A,B , then the following inequality holds: det(A+B)^\frac{1}{n} \geq det(A) ^\frac{1}{n} + det(B)^\frac{1}{n} Homework Equations Brunn-Minkowski Inequality...

Let $$A$$ be the $$n\times n$$ matrix: \begin{equation} A= \begin{bmatrix} % or pmatrix or bmatrix or Bmatrix or ... 2 & 1&\dots & 1 & 1 \\ 1 & 2&\dots & 1 & 1 \\ \vdots&\ddots & \ddots & 2 & 1 \\ 1 & & \dots & 1&2 \\ \end{bmatrix} \end{equation}...

Homework Statement Hello, I am stuck on this particular question for my homework. It is a 4x4 Matrix that consists of a b b b b a b b b b a b b b b b The Attempt at a Solution My approach has been to factor out the a to give the first row of 1 b b b and then use...

Hi, I am working through Chapter 4 of Francesco, Mathieu and Senechal's CFT book (https://www.amazon.com/dp/038794785X/?tag=pfamazon01-20). Equation 4.52 states that for a special conformal transformation \left|\frac{\partial\textbf{x'}}{\partial\textbf{x}}\right| =...

I have been asked to prove that the determinant of any matrix representing a Lorentz transformation is plus or minus 1. I can see that the determinant of the Lorentz transformation matrix is 1, but don't know how to prove +-1 in general. How to generalise the lorentz transformation? I've also...

I have read that when 2 Hydrogen atoms come together their individual spatial wavefunctions overlap in the following way: ψsymmetric = ψa + ψb ... bonding case ψasymmetric = ψa - ψb ... antibonding case How do you express this in terms of the Slater Determinant?

This is what the symbols in the question represent( sorry about the syntax) ; sr = s subscript r a^r = alpha to the power of r b^r = beta to the power of r g^r = gamma to the power of r Question: If sr = a^r + b^r + c^r, by expressing the determinant as the product of two determinants...

Hello, I'd like to know if the following two paragraphs regarding the determinant of a matrix are correct and also, am I missing any other important implications by calculating the determinant? any other important things I can find from with that value? thanks. 1. If det A=0 <=> Linear...

Hey people, could someone solve this problem and explain step by step? It's from a past exame and I really need to know how to do it, tried everything I know (not much tho). Please don't omit steps.Really appreciated! imageshack.us/photo/my-images/403/dvidafrumdelgebra.png/

Homework Statement I've attached the problem, it involves reducing a 3x3 matrix determinant to row echelon form, but the leading diagonal elements have to be linear in a and b afterwards. Homework Equations The Attempt at a Solution I've managed to convert it to row echelon form...

Let \begin{equation*} A=% \begin{bmatrix} 0 & 1 & \cdots & n-1 & n \\ 1 & 0 & \cdots & n-2 & n-1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ n-1 & n-2 & \cdots & 0 & 1 \\ n& n-2 & \cdots & 1 & 0% \end{bmatrix}% \end{equation*}. How can you prove that det(A)=[(-1)^n][n][2^(n-1)]? Thanks.

How to prove that the determinant of a symmetric matrix with the main diagonal elements zero and all other elements positive is not zero and different ?

Hello, I have the following determinant: \text{det}\left(\mathbf{A}\mathbf{A}^H\right) where H denoted complex conjugate transpose, and A is a circulant matrix. I am looking for a lower bound for the above determinant. Is there one? Thanks in advance

Homework Statement For the function f(x, y) = xye^[-(x^2 + y^2)] find all the critical points and classify them each as a relative maximum, a relative minimum, or a saddle point. Homework Equations Partial differentiation and Hessian determinants. The Attempt at a Solution I get how...

Homework Statement Homework Equations The Attempt at a Solution I tried to see if the problem has any properties with determinants that i can apply but the properties i learned didn't involve the use of adjoint matrices so I'm kind of stumped on this one. Any hints would be...

Any help would be appreciated. I tried to solve this problem by first the adjoint of A but then that get really complicated I have no clue how to do this.

Hi. This is a property but I got confused with it when I thought about the quotient. Look at this: \left| {\begin{array}{*{20}{c}} a&d&{2g}\\ {\frac{1}{3} \cdot 3b}&{\frac{1}{3} \cdot 3e}&{\frac{1}{3} \cdot 6h}\\ c&f&{2i} \end{array}} \right| = \frac{1}{3} \cdot \left|...

In "A Short Course in General Relativity", I met a statement that says if we denote the value of the determinant |g_{ab}| by ##g##, then the cofactor of ##g_{ab}## in this determinant is ##gg^{ab}## and following this we can deduce ∂##_cg=####(##∂##_cg_{ab})gg^{ab}##. First, I don't...

Homework Statement Let A be the matrix with eigenvalues x1 = 2, x2 = 1, x3 = 1/2 , x4 = 10 and corresponding eigenvectors v1: <1,-1,1,0>, v2: <1,-1,0,0>, v3: <1,0,0,1>, v4: <0,0,1,1> Calculate |A|Homework Equations See aboveThe Attempt at a Solution I'm not really sure how to start this...

Homework Statement 1 1 1 a b c = (b-a)(c-a)(c-b) a^2 b^2 c^2 (above is a 3x3 matrix equaling to a equation) question:"Show by applying property of the determinant"Homework Equations N/AThe Attempt at a Solution read through the whole chapter of...

please look at the attachement and my attempt at the solution - hope you can help. Thanks

Homework Statement If U is a 2 x 2 unitary matrix with detU=1. Show that |TrU|≤2. Write down the explicit form ofU when TrU=±2 Homework Equations Not aware of any particular equations other than the definition of the determinant and trace. The Attempt at a Solution I have...

1. Construct 6x6, 5x5, 4x4 and 3x3 matrices that has the largest determinant possible using only 1 and -1 I have attempted to reduce this problem by applying determinant properties, here is an example of my work for a 3x3 matrix. If we have 3x3 and fill it with 1's and -1's the total...

So I suck at programming, but I need to find the determinant of a complex 6x6 array using GSL in C (not GSL complex, complex.h complex). Here is what has failed so far starting with a 6x6 double complex array named mymatrix: gsl_matrix_complex_view m = gsl_matrix_complex_view_array(mymatrix, 6...

\[ \text{Let u,v } \in \mathbb{R}^n \;\; \text{Show that } \;\;, Det(I + uv^T) = 1 + v^T u \] I is the identity matrix nxn any hints ?

What exactly does it mean when the determinant of a Jacobian matrix vanishes? Does that imply that the coordinate transformation is not a good one? How do you know if you coordinate transformation is a good one or a bad one?

Hey libniz formula i can't post picture and i can't use latex :( if we want to use Libniz forumla for find the determinant of A such that A = 1..-1..0..3 ...2..-3..0..4 ...0..2..-1..5 ...1..2..2..3 sigma here from S_4 which has 4! elements which element will we choose

Dear all, I have a simple question for you. Any help will be very appreciated. ==Assumptions=== I have a 2x2 matrix, with real entries: |A B| |C D| Such matrix has unit determinant, AD-BC=1. (For those of you in group theory: the above is a representative of SL(2,R))...

I need help with another homework problem Let n be a positive integer and An*n a matrix such that det(A+B)=det(B) for all Bn*n. Show that A=0 Hint: prove property continues to hold if A is modified by any finite number of row or column elementary operations It seems obvious that A=0 but...

I'm having trouble with this problem on my homework Let n be a positive integer and A=[ai,j] A is n*n. Let B=[Bi,j] B is n*n be the matrix defined by bi,j=(-1)i+j+1 for 1<i,j<n. Show that det(B)=(-1)ndet(A) Hint: use the definition of determinant I honestly have no idea how to go about...

how we can find the determinant of non square matrix ??

Homework Statement In R^2, vectors x = (x1, x2) and a = (a1, a2). For fixed a, det(a, x) is a scalar-valued linear function of the vector x. Thus it can be written as the dot product of x with some fixed vector w. Explain why w is perpendicular to a. Do not use an expression of w in terms of...

Homework Statement What proportion of 2x2 symmetric matrices with entries belonging to [0, 1] have a positive determinant? Homework Equations A^{T} = A If A = [[a, b], [c, d]] Then det(A) = ad - bc. But A is symmetric, so c = b. So det(A) = ad - b^2 So, in order for A to have a...