1. ### Using a determinant to find the area of the triangle (deriving the formula)

This is the question. The following is the solutions I found: I understand that the first line was derived by setting one vertex on origin and taking the transpose of the matrix. However, I cannot understand where the extra row and column came from in the second line. Can anyone explain how...
2. ### Finding the Determinant to find out if the matrix is invertible

question: My first attempt: my second attempt: So I am getting 0 (the right answer) for the first method and 40 for the second method. According to the theorem, shouldn't the determinant of the matrix remain the same when the multiple of one row is added to another row? Can anyone explain...
3. ### I Meaning of this Levi-Civita Contraction?

Hi all, I'm doing undergraduate research this summer, and a few times I've been told to calculate a term with the following form: ∈abcdpaqbkcsd, where p,q,k and s are four vectors (four-momentum, spin, etc). Now I know this ends up calculating exactly like a 4x4 determinant, I'm just not quite...
4. ### Problem involving complex numbers

Homework Statement Refer given image. Homework Equations Expansion of determinant. w^2+w+1=0 where w is cube root of 1. The Attempt at a Solution Expanding the determinant I got cw^2+bw+a-c=0. Well after that I have no idea how to proceed.
5. ### Time Derivative of Rank 2 Tensor Determinant

Homework Statement Show that for a second order cartesian tensor A, assumed invertible and dependent on t, the following holds: ## \frac{d}{dt} det(A) = det(a) Tr(A^{-1}\frac{dA}{dt}) ## Homework Equations ## det(a) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} A_{il}A_{jm}A_{kn} ## The...
6. ### I Determinant problem in an article about QCD phase diagram

Hi. I'm reading an article about QCD phase diagram. https://arxiv.org/abs/1005.4814. I want to derive eq(20), but I don't know how. Does anyone know how to derive this?
7. ### Expressing difference product using Vandermonde determinant.

Homework Statement Show that ##g=g(x_1,x_2,...,x_n)=(-1)^{n}V_{n-1}(x)## where ##g(x_i)=\prod_{i<j} (x_i-x_j)##, ##x=x_n## and ##V_{n-1}## is the Vandermonde determinant defined by ##V_{n-1}(x)=\begin{vmatrix} 1 & 1 & ... & 1 & 1 \\ x_1 & x_2 & ... & x_{n-1} & x_n \\ {x_1}^2 & {x_2}^2 & ... &...
8. ### Proof regarding determinant of block matrices

Homework Statement Let A,B,C,D be commuting n-square matrices. Consider the 2n-square block matrix ##M= \begin{bmatrix} A & B \\ C & D \\ \end{bmatrix}##. Prove that ##\left | M \right |=\left | A \right |\left | D \right |-\left | B \right |\left | C \right |##. Show that the result may not be...
9. ### I Using determinant to find constraints on equation

Basically I don't know how to get to the constraints from the system of equations. In class we used det to find constraints for homogenous equations, but we didn't go over this situation. Someone spell it out for me?
10. ### Using eigenvalues to get determinant of an inverse matrix

Homework Statement Homework Equations determinant is the product of the eigenvalues... so -1.1*2.3 = -2.53 det(a−1) = 1 / det(A), = (1/-2.53) =-.3952 The Attempt at a Solution If it's asking for a quality of its inverse, it must be invertible. I did what I showed above, but my answer was...
11. ### Determinant of a 3x3 matrix via row reduction

Homework Statement Show that the determinant of is (a-b)(b-c)(c-a) Homework Equations Row reduction, determinants The Attempt at a Solution Apparently I got a (a-b)^2 instead of (a-b) when I multiplied them up. It would be grateful if someone can point me out where the mistakes are.
12. ### A Exterior Algebra Dual

The determinant of some rank 2 tensor can be expressed via the exterior product. $$T = \sum \mathbf{v}_i \otimes \mathbf{e}_i \;\;\; \text{or}\sum \mathbf{v}_i \otimes \mathbf{e}^T_i$$ $$\mathbf{v}_1\wedge \dots \wedge \mathbf{v}_N = det(T) \;\mathbf{e}_1\wedge \dots \wedge\mathbf{e}_N$$ The...
13. ### I Determinant of A^t A

I have a problem of proving an identity about determinants. For ##A\in M_{m\times n}(\mathbb{R}),## a matrix with ##m## rows and ##n## columns, prove the following identity. $$|\det(A^tA)|=\sum_{1\le j_1\le ... \le j_n \le m} (det(A_{j_1...j_n}))^2$$ where ##A_{j_1...j_n}## is the matrix whose...
14. ### B Application of Matrices and Determinants

Hello I was learning about determinants and matrices. I learned the generalization of getting the determinant of an n by n matrix. I then applied this to vector space (i + j + k) via a cross product and noticed that you leave the i j and k in their own columns in the first row of the matrix...
15. ### Determinant of 3x3 matrix equal to scalar triple product?

The determinant of a 3x3 matrix can be interpreted as the volume of a parallellepiped made up by the column vectors (well, could also be the row vectors but here I am using the columns), which is also the scalar triple product. I want to show that: ##det A \overset{!}{=} a_1 \cdot (a_2 \times...
16. ### Having trouble solving using properties of determinants ...

Homework Statement I'm a bit at a loss - I thought the last row with '1's would be useful, but it just gave me: (b2c - bc2) - (a2c - ac2) + (a2b - ab2) and bc(b - c) - ac(a - c) + ab(a - b) But then it is a dead end. I am probably doing something stupid again ... Any help appreciated.
17. ### Evaluate the partial derivative of a matrix element

Homework Statement A determinant a is defined in the following manner ar * Ak = Σns=1 ars Aks = δkr a , where a=det(aij), ar , Ak , are rows of the coefficient matrix and cofactor matrix respectively. The second term in the equation is the expansion over the columns of both matrices, δkr is...
18. ### Insight into determinants and certain line integrals

I just did this following exercise in my text If C is the line segment connecting the point (x_1,y_1) to (x_2,y_2), show that \int_C xdy - ydx = x_1y_2 - x_2y_1 I did, and I also noticed that if we put those points into a matrix with the first column (x_1,y_1) and the second column (x_2,y_2)...
19. ### Solving Singular matrices

Hi, How would you solve a singular matrix? ie when determinant is zero. Lets assume an equation, (1) Ax+By=E and (2) Cx+Dy=F if the determinant; AD-BC = 0, and therefore the matrix is singular, How to go around solving the equation? LU decomposition, Gaussian elimination? Ideally I am...
20. ### Pfaffian and determinants of skew symmetric matrices

Can anyone explain or point me to a good resource to understand these operators? I'm trying to the understand determinants for skew symmetric matrices, more specifically the Moore determinant and it's polarization of mixed determinants. Can hone shed some light? I'm confused as to how the...
21. ### Help with Wronskian Equation

Homework Statement W(t) = W(y1, y2) find the Wronskian. Equation for both y1 and y2: 81y'' + 90y' - 11y = 0 y1(0) = 1 y1'(0) = 0 Calculated y1: (1/12)e^(-11/9 t) + (11/12)e^(1/9 t) y2(0) = 0 y2'(0) = 1 Calculated y2: (-3/4)e^(-11/9 t) + (3/4)e^(1/9 t) Homework Equations W(y1, y2) = |y1...