Determinant Definition and 494 Threads

Homework Statement The matrix A is given as: [25 , 5] [-150 , -30] find, in terms of m, the expression det(A-mI). This will be in a quadratic expression in m. Homework Equations det(A) = ad-bc The Attempt at a Solution i have no idea how to approach this but i...

Assume that X^1,X^2,\ldots, X^k are vectors in \mathbb{R}^n, and 1\leq k\leq n. Is there a simple formula for the k-dimensional measure of the generalised "quadrangle" spanned by these vectors? If k=n, then the solution is |\textrm{det}(X)| with X_{ij}=(X^{j})_i. If k=2 and n=3, then the...

If given a n*n matrix with all rows and columns sum to 0, how do I argue that all its (n-1)*(n-1) minor have the same determinant up to a sign? Since all rows and columns all sum to 0, then I know that any column is a linear combination of all others, so that the determinant of this n*n matrix...

Find the determinant of the following 4x4 matrix -1 0 0 0 -15 7 -15 5 -5 1 -6 1 6 -7 6 -5 Here is my attempt Ive seen that I should do the following: -1.(determinant of 3x3 matrix)-0.(determinant of 3x3 matrix) + 0.(determinant of 3x3 matrix)-0.(determinant of...

FInd the determinant of the following matrix? 4,-4,-8 -2, 2, 6 0, 0,-1 Heres my attempt 4.(2x(-1)- 6x0) -(-4).((-2)x(-1) - 6x0) +(-8).((-2)x0-2x0) which goves: 4.(-2)+4(2) -8 = 0 is this correct?? Im also asked to find the trace? What is this and how do i find it...

Hello everybody, To compute an observable in a physical problem I need to compute the determinant of a skew matrix (http://en.wikipedia.org/wiki/Skew-symmetric_matrix#Determinant"). The straightforward way to numerically calculate this determinant is to use a lapack routine that compute the...

So. I've been told by my prof that the best way to find the determinant of a matrix is to row reduce it to upper triangular and then take the product of the numbers on the diagonal. That's fine, BUT, how do you know how to reduce it? Depending on what row operations you do, you get...

Homework Statement A = \left(\begin{array}{cccc} 3 & 5 & -2 & 7\\ 1 & 2 & -1 & 1\\ 2 & 4 & 1 & 5\\ 3 & 7 & 5 & 3 \end{array} \right)Homework Equations The Attempt at a Solution R4=R4-R1 R3=R3-2R2 A = \left(\begin{array}{cccc} 3 & 5 & -2 & 7\\ 1 & 2 & -1 & 1\\ 0 & 0 & 3 & 3\\ 0 & 2 & 7 & -4...

Homework Statement Let A be a a square n*n matrix. Prove that A^-1 has only integer enteries if and only if the determinant of A is + or -1. Homework Equations general knowledge of determinants The Attempt at a Solution Proof: => Suppose that det(A) = 1 (without losing...

Homework Statement So, I'm in a 2D Euclidean space V. The two vectors v1 and v2 forms the orthonormal basis set for V. Consider the transformation T(v1) = a*v1 + b*v2, T(v2) = c*v1 + d*v2 Show, using scalar products (not vector products), that the area scale factor of the transformation T is...

Homework Statement The probem is to show within 10 seconds that Det(\vec{V} \otimes\vec{V})=0 with \vec{V}(x,y,z)=(Vx(\vec{r}),Vy(\vec{r}),Vz(\vec{r})) The Attempt at a Solution So I thought if I show the matrix is not invertible I would be done. But even with that I can't easily solve...

Homework Statement Let A,B ε Mnxn(C) Prove that if B is invertible, then there exists a scalar cεC s.t. A + cB is not invertible. HINT: examine det (A+cB) Homework Equations The Attempt at a Solution I know that the det(A+cB)=0 since a non invertible matrix has det=0 I know...

Homework Statement A=(r1, r2, r3)transpose and detA=2. Let B=(2r1+3r2, r3+2r1, r2-r1-3r3). Find detB. Homework Equations - for row switch the det is multiplied by -1 - if you add a row to another it has no effect on det - if you multiply a row by a constant k the det will be k*detThe...

Homework Statement I am finding the determinant by first reducing to row echelon form. Please help I have been agonizing over this lol Homework Equations I have got to this stage and am now going to do 3rd row - 4th row and replace the 4th row as you do to get to row echelon. At this...

Homework Statement I think I have broken maths. I am reducing a matrix to row echelon form to find the determinant. The matrix I will show is nearly in the desired form Homework Equations 1 -3 -2 1 0 1 2 -1 0 0 1 -0.8 0 0 1 0 The Attempt at a Solution As it stands...

Hi, Given two real-valued positive-definite matrices A and B, assume one is greater than the other with respect to positive definite ordering. That is, A>B. Does the following implication hold? A>B \Rightarrow \text{det}(A)>\text{det}(B) Thanks.

Solve the following equation involving a determinant det x (x+2) 9 2x = 0 I did x * 2x - 9(x+2) It came out to 2x^2 - 9x - 18 = 0 Do I have to do anything else like a quadratic equation? I went to a math tutor and he wasn't sure. I scanned the problem and it's number 11...

Homework Statement Evaluate the following determinant by expressing the matrix in upper triangular form. Homework Equations row operations. The Attempt at a Solution http://yfrog.com/jnscan0001ugj http://img707.imageshack.us/img707/6379/scan0001ug.jpg...

Homework Statement well, my assignment was to make a gauss elimination, so now i need to compute the determinant of an n by n diagonal matrix variable rows = number of equations variable i = random integer matrix A[100][100] dummy matrix A2[100][100] Homework Equations det[A]...

Homework Statement matrix A= [v1,v2,v3,v4]^T and detA=4 then what is det [9v1+7v4, v2, v3, 9v4+4v1]^T ? Homework Equations None. The Attempt at a Solution I know that my problem is identical to the problem encountered in https://www.physicsforums.com/showthread.php?t=431395"...

Homework Statement the problem is to prove that determinant(AB)=det(A)det(B) Homework Equations i don't think there is any equation. :( The Attempt at a Solutioni can figure it out by taking arbitrary elements in rows and columns , but i was wondering if i can prove it in a...

Homework Statement Consider the n x n determinant in which each entry on the main diagonal is a 2, each entry on the two adjacent diagonals is a 1, and every other entry is a zero. Expand along the first row to show that B_n=2B_n-1 - B_n-2 Homework Equations n/a The Attempt at a...

Homework Statement The matrix is in the following form 2 4 -2 -2 1 3 1 2 1 3 1 3 -1 2 1 2 Homework Equations The Attempt at a Solution I subtracted equation 2 from equation 3 and came up with the following matrix 2 4 -2 -1 1 3 1 2 0 0 0 1 1 -2 1 2 It seems to make...

Homework Statement Let E be a vector space of finite dimension over \Gamma (char 0), and F\colon L(E,E)\rightarrow \Gamma satisfies (1) F(\phi \circ \psi)=F(\phi)F(\psi) (2) F(\hbox{id})=1 Prove F can be expressed as a function of determinant, F(\phi)=f(\hbox{det}\phi).Homework Equations...

Homework Statement The 4x4 matrix with rows v1, v2, v3 and v4 has a determinant of -5. What is the determinant of the matrix with rows v1, v2, 7v3+6v4, 6v3+8v4? Homework Equations The Attempt at a Solution I tried doing -5x7x8=-280 but its saying its wrong. I don't understand...

Problem: Show, without evaluating directly, that \left|\begin{matrix} a_1+b_1t&a_2+b_2t&a_3+b_3t \\ a_1t+b_1&a_2t+b_2&a_3t+b_3 \\ c_1&c_2&c_3 \end{matrix}\right| = (1-t^2)\left|\begin{matrix} a_1&a_2&a_3 \\ b_1&b_2&b_3 \\ c_1&c_2&c_3 \end{matrix}\right| Clearly, here I'm...

Hello, I am supposed to prove that the determinant of a second order tensor (a matrix) is equal to the following: det[A] = \frac{1}{6} \epsilon_{ijk} \epsilon_{pqr} A_{pi} A_{qj} A_{rk} anyone have any idea how i would go about this? any method is welcome where the determinant of the...

Hello, I am supposed to prove that the determinant of a second order tensor (a matrix) is equal to the following: det[A] = \frac{1}{6} \epsilon_{ijk} \epsilon_{pqr} A_{pi} A_{qj} A_{rk} anyone have any idea how i would go about this? any method is welcome

There is a small outline in a book about finding the determinant of a matrix over an arbitrary commutative ring. There are a few things I don't understand; here it is: 'Let R be a commutative ring with a subring K which is a field. We consider the matrix X = (x_{ij}) whose entries are...

Homework Statement Prove: A 2x2 matrix is nonsingular if and only if the determinant != 0 The Attempt at a Solution I need to prove this, using logic and maybe the theorem that a n x n matrix is nonsingular if and only if it is row equivalent to I_n. I could use a push in the...

Hello everyone, So, I read somewhere that the Jacobian determinant of a transformation determines the local volume change. Say I am in 3D space and I have the following relationship: F(x', y', z') = F(x, y, z) + T(x, y, z) The LHS gives the new position and the RHS is the old position + the...

Homework Statement Row-1...a 1 1 Row-2...1 b 1 Row-3...1 1 c if the value of the determinant is positive the what is abc? Homework Equations ans is abc>-8 The Attempt at a Solution I am not able to follow the method shown in R D sharma Book. Is there any other method to...

Homework Statement I have a determinant to solve. Fistrow is -2a, a + b, a+c Second row is b + a, -2b, b + c Third row is c+a, c+b, -2c Prove that the determinant is equal to 4 (b +c) (c +a) (a + b) Homework Equations Not applicable. The Attempt at a Solution I have...

Let X and Y be n x n real matrices such that the 2n x 2n matrix M:= X -Y Y X is invertible. What is the relationship between det(M) and det(X+iY)??

I have a midterm tomorrow and I find I'm quite slow at finding the determinant of a 3x3 matrix. Basically I'll only need to find the determinant to find the characteristic polynomial (at least for this class) and my prof on the board does it so fast, I'm wondering if there's some trick I missed...

Hope someone can help me out here.. I've found the eigenvalues (lamda) of this matrix, but through a very very long way, does anyone know of a quicker way (there must be a quicker way). The matrix is 3x3: H = ( E, a, a; a, E, a; a, a, E ). I can reduce the determinant to the following, but...

How could one derive a definition of the determinant from some of its basic properties such as det of product = product of dets or the determinant of a transpose is the determinant of the untransposed matrix? Upon instigating research on determiniants, all I've found are definitions that...

Hi... We have all seen the equation det(M)=exp(tr(lnM)). I was taught the proof using diagonalisation. I was wondering if there was a proof for non-diagonalisable matrices also.

Hi; I am pretty sure that sqrt(-g) is diffeomorphism-invariant. I am wondering if all powers of this are diffeo-invariant too. For example, are -g, g^2, etc. all invariants too?

Use the trace and determinant to compute eigenvalues. I know how to do this with a 2x2 but not sure how to do it with a matrix of nxn where n>2. \begin{bmatrix} \frac{1}{2} & \frac{1}{3} & \frac{1}{5}\\ \frac{1}{4} & \frac{1}{3} & \frac{2}{5}\\ \frac{1}{4} & \frac{1}{3} & \frac{2}{5}...

"Determinant" of a non-square matrix? Hi, is there any numerical invariant that would characterize the rank of a non-square matrix, similar to the determinant for square matrices? I.e. having a matrix nxm, with n<m, I'm looking for a number that would be zero if the rank of the matrix is...

How can we find the determinant of a complex matrix?

Is there a good method or any particular way you guys use to factorise determinant to avoid staring at a problem for minutes without coming up with any answers? Lets say I want to factorize \begin{bmatrix} 1 & a & a+1 \\ a+1 & 1 & a \\ a & a+1 & 1 \end{bmatrix} Where should I start...

Did someone just realize that taking the determinant of a specific matrix gives you the cross product formula, or is there is a specific conceptual reason why it works?

Homework Statement Linear transformation T: P2->P2 T(f) = -5f + 8f' Need to find detA (A is a matrix of T) Homework Equations T(f) = Af The Attempt at a Solution The basis of P2 is B={1, x, x2}. Some polynomial f with respect to B looks like this in general: (a, b, c)T right...

Hi, I've been trying to get my head around this. \Sigma_{(j)} is a p x p matrix given by \Sigma_{(j)} = \left(\begin{array}{cc}\sigma_{jj} & \boldsymbol{\sigma_{(j)}'}\\\boldsymbol{\sigma_{(j)}} & \boldsymbol{\Sigma_{(2)}}\end{array}\right) where \sigma_{jj} is a scalar...

Hi, Is there a simplification for the determinant of a symmetric matrix? For example, I need to find the roots of \det [A(x)] where A(x) = \[ \left( \begin{array}{ccc} f(x) & a_{12}(x) & a_{13}(x) \\ a_{12}(x) & f(x) & a_{23}(x) \\ a_{13}(x) & a_{23}(x) & f(x) \end{array}...

I am reading through David Widder's Advanced Calculus and he abbreviates a determinant as: \left( \begin{array}{cccc} r_{1} \ s_{1} \ t_{1}\\ r_{2} \ s_{2} \ t_{2}\\ r_{3} \ s_{3} \ t_{3}\\ \end{array} \right) And refers to it by (rst). He then states...

Hi all, I'm studying my mathematics lesson, and there is an example I can't understand: Consider the matrix A=(0 In) (-In 0) with In the identity nxn We want to compute detA : We introduce the permutation p=(1 2 ... n n+1 ... 2n)...

Homework Statement Find the determinant of the following matrix: 4...0...1 19...1...-3 7...1...0 I chose the 1st row to do the operations on. 4 [(1*0) - 1*(-3)] + 1 [19*1 - 7*1] = 4[0 - (-3)] + 1[12] =12 + 12 =24 I can't see any mistakes in that, but it's...