Determinant Definition and 494 Threads

$\textsf{Compute the determinant of} $ $$A=\left| \begin{array}{rrr} 1&0&2\\ 1&0&0 \\ 3&2&0\end{array} \right|$$ $\textsf{(a)by method of Basket weaving}$ $\begin{array}{rrrrr} 1&0&2&1&0 \\ 1&0&0&1&0\\ 3&2&0&3&2 \end{array}$ $[(1)(0)(0)+(0)(0)(3)+(2)(1)(2)]-[(3)(0)(2)+(2)(0)(1)+(0)(1)(0)]$...

$\textsf{a. Find the determinants by row reduction in echelon form.}$ $$\left| \begin{array}{rrr} 1&5&-6\\ -1&-4&4 \\ -2&-7 & 9 \end{array} \right|$$ ok i multiplied $r_1$ by 1 and added it to $r_2$ to get $$\left| \begin{array}{rrr} 1&5&-6\\ 0&1&-2 \\ -2&-7 & 9 \end{array} \right|$$...

Hey! :o Let $\alpha, \beta, \gamma$ be internal angles of an arbitrary triangle. I want to show that $$\det \begin{pmatrix}\cos \beta & \cos \alpha&-1 \\ \cos\gamma & -1 & \cos\alpha \\ -1 & \cos\gamma & \cos\beta\end{pmatrix}=0$$ We have the following: \begin{align*}&\det...

Hey! :o Let $A \in\mathbb{R}^{n\times n}$, $n\geq 3$ be a matrix with $n+1$ elements $1$ and the remaining elements are $0$. I want to show that $\det (A)\in \{-1, 0, 1\}$ and each of these $3$ possible values can occur. Could you give me a hint how we could show that? I got stuck right now...

I’m hoping to clear up some confusion I have over what the Lie derivative of a metric determinant is. Consider a 4-dimensional (pseudo-) Riemannian manifold, with metric ##g_{\mu\nu}##. The determinant of this metric is given by ##g:=\text{det}(g_{\mu\nu})##. Given this, now consider the...

Homework Statement Homework EquationsThe Attempt at a Solution [/B] Det( ## e^A ## ) = ## e^{(trace A)} ## ## trace(A) = trace( SAS^{-1}) = 0 ## as trace is similiarity invariant. Det( ## e^A ## ) = 1 The answer is option (a). Is this correct? But in the question, it is not...

Homework Statement \begin{vmatrix} 1 & -4 & 3 & 4 \\ 0 & -9 & 6 & 8 \\ 0 & -6 & 5 & 5 \\ 0 & 0 & -3 & 2 \end{vmatrix} so the determinant of this matrix is -9, apparently I am doing something illegal in my row operations. I want to get -6 in row 3 to be 0 so... 2R2 - 3R3 = 0 0 -3 1...

Homework Statement In special relativity the metric is invariant under lorentz transformations and therefore so is the determinant of the metric. How does the metric determinant transform under a more general transformation $$x^{a\prime}=J^{a\prime}_{\quad a}x^{a}$$ where $$J^{a\prime}_{\quad...

This is something I seek a proof of. Theorem: Let ## \mbox{det}:\mbox{Mat}_{n\times n}(\mathbb{R}) \rightarrow \mathbb{R}## be the determinant function assigned to a general nxn matrix with real entries. Prove this mapping is continuous. My attempt. Continuity must be judged in...

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?

Hey! :o We have $3$ lines with equations $a_{i1}x+a_{i2}y+a_{i3}=0$, $i=1,2,3$. I want to show that $\det ((a_{ij}))=0$ iff the lines are pairwise parallel of they have a common point. We have that $\det ((a_{ij}))=0$ iff we have a zero row. That would mean that we have linear independency...

Hello there, Suppose $f$ smoothly maps a domain ##U## of ##\mathbb{R}^2## into ##\mathbb{R}^3## by the formula ##f(x,y) = (x,y,F(x,y))##. We know that ##M = f(U)## is a smooth manifold if ##U## is open in ##\mathbb{R}^2##. Now I want to find the determinant of the metric in order to compute the...

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 & ... &...

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...

Hey! :o Suppose we have the matrix $A_n=(A_{ij})\in \mathbb{C}^{n\times n}$ with $a_{ij}=\left\{\begin{matrix} 1 , & i=j\\ -1 , & i=j-1\\ j^2, & i=j+1\\ 0 , & \text{ otherwise} \end{matrix}\right.$ for $1\leq i,j\leq n$. I want to find the determinant using induction. I have done the...

Homework Statement Show $$\frac{\partial \det(A)}{\partial A_{pq}} = \frac{1}{2}\epsilon_{pjk}\epsilon_{qmn}A_{jm}A_{kn}$$ Homework Equations ##\det(A)=\epsilon_{ijk}A_{1i}A_{2j}A_{3k}## The Attempt at a Solution $$\frac{\partial \det(A)}{\partial A_{pq}}=\frac{\partial}{\partial...

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?

I'm a little stuck here. I should determine the following determinant. I first tried to simplify it a little by using elemntary transformations. And then I did Laplace expansion on the last row. $\begin{vmatrix}2 & 2 & \cdots & 2 & 2 & 1 \\ 2 & 2 & \cdots & 2 & 2 & 2 \\ 2 & 2 & \cdots & 3 & 2 &...

I was given this $n \times n$ matrix $A$ which is a mirror-image of identity matrix, ie., its non-main diagonal consists of entries of $1$, the rest of entries are $0$. I need to find out the determinant of $A$. Having experimented with $n = 2, 3, ...,$ I observed that for $n = 2 + 4k$ or $n = 3...

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...

Given a n x n matrix whose (i,j)-th entry is i or j, whichever smaller, eg. [1, 1, 1, 1] [1, 2, 2, 2] [1, 2, 3, 3] [1, 2, 3, 4] The determinant of any such matrix is 1. How do I prove this? Tried induction but the assumption would only help me to compute the term for Ann mirror.

The problem I have is this: Show that \begin{bmatrix} 1 & 1 & 1 \\ λ_{1} & λ_{2} & λ_{3} \\ λ_{1}^{2} & λ_{2}^{2} & λ_{3}^{2} \end{bmatrix} Has determinant $$ (λ_{3} - λ_{2}) (λ_{3} - λ_{1}) (λ_{2} - λ_{1}) $$ And generalize to the NxN case (proof not needed)Obviously solving the 3x3 was...

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.

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...

Homework Statement Hi there, I'm happy with the proof that any odd ordered matrix's determinant is equal to zero. However, I am failing to see how it can be done specifically for a 3x3 matrix using only row and column interchanging. Homework Equations I have attached the determinant as an...

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...

Homework Statement Using index notation only (i.e. don't expand any sums) show that: \begin{align*} &\text{(a) } \epsilon_{ijk} \det \underline{\bf{A}} = \epsilon_{mnp} A_{mi} A_{nj} A_{pk} \\ & \text{(b) } \det \underline{\bf{A}} = \epsilon_{mnp} A_{m1} A_{n2} A_{p3} \end{align*} Homework...

This property is given in my book. The square of any determinant is a symmetric determinant. Well it works when I take a determinant say 3x3 and multiply it by itself using row to row multiplication. But it fails if I multiply using row to column. Thanks

Hello, Could someone provide me with a good proof or explain me here how we can derive Slater determinant for N fermions by starting with the vacuum state and the creation operators with anticommutation equations. I see that the idea of both these structures is similar but I cannot work it out...

I know row reduction methods are the best way to calculate the determinant of large matrices. I was wondering if you can use the appended matrix method to calculate the determinant of a 4x4 by appending the matrix with the first 3 columns. There should be n! terms, but I only get 8 instead of 24.

Ok, I've got these functions to get the x (right), y (up) and z (forward) coordinates to plot with my computer program: x = r*Math.cos(a)*Math.sin(o) y = r*Math.sin(a) z = -r*Math.cos(a)*Math.cos(o) It's the equations of a sphere where I've placed the origin (o,a,r) = (0,0,0) of the source...

Hi all, I have a question about Slater Determinant for excited states. Let's say we want to construct approximate (2 level) wavefunction of He in some certain state. Since we have two electrons in two level system with spin in consideration, we can construct total of 4 different wavefunctions...

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...

Homework Statement I have to solve the following determinant ## D_n=\begin{vmatrix} 1 & 1 & 1 & \cdots & 1 & 1 & 1 \\ 1 & 1 & 1 & \cdots & 1 & 2 & 1 \\ 1 & 1 & 1 & \cdots & 2 & 1 & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 1 & 2 & 1 & \cdots & 1 & 1 & 1 \\ 1 & 1 & 1 &...

Hi, I've just wierded myself out so time to stop for today, but afore I go ... Angular momentum matrices (also Pauli) anti-commute as $ [J_x, J_y] = iJ_z $ So $ Det\left( [J_x, J_y] \right) = i Det(J_z) $ $\therefore Det(J_xJ_y)-Det(J_yJ_x) = i Det(J_z) $ $\therefore...

Making sure I have this right, $ |A| = \sum_{i}\sum_{j}\sum_{k} \epsilon_{ijk}a_{1i}a_{2j}a_{3k} $ (for a 3 X 3) and a 4 X 4 would be $ |A| = \sum_{i}\sum_{j}\sum_{k} \sum_{l} \epsilon_{ijkl} a_{1i} a_{2j} a_{3k} a_{4l} $ ? Is there any special algebra for these terms? (they could be...

Find the general form of an orthogonal 2 x 2 matrix = $ \begin{bmatrix}a&b\\c&d\end{bmatrix}$ I used $det(A)=\pm 1$ (from an earlier exercise) - and the special form of $ A_{2 \times 2}^{-1} = \frac{ \begin{bmatrix}d&-b\\-c&a\end{bmatrix}}{|A|} $ using $|A| = +1$ first, to get an...

Probably trivial, but for matrices with different ranks, Det is not closed for addition? I think it is closed under multiplication? So really I must show Det not closed under addition for square matrices of the same order... $ D(A_n) = \sum_{j=1}^{n} a_{1j}C_{ij} $ and $ D(B_n) =...

Homework Statement I'm trying to calculate the variation of the following term for the determinant of the metric in the polyakov action: $$h = det(h_{ab}) = \frac{1}{3!}\epsilon^{abc}\epsilon^{xyz}h_{ax}h_{by}h_{cz}$$ I know that there are some other ways to derive the variation of a metric...

1. Given A,B\in Mat _n(\mathbb{R}) 2. Show that: a) \det (A^2 + A + E)\geq 0 b) \det (E+A+B+A^2+B^2)\geq 0 , where E is the unit matrix.3. My attempt at a solution A^2 + A + E=(A + E)^2 -2A https://drive.google.com/file/d/0B8zKPTh1siSsOHNWQnBfaXR3QXM/view?usp=sharing pleas give me tips to solve

In HS they just taught you the formula for it (the cofactor method) and a few other things, such as det(A) = 0 means A is singular. I finally reached Ch 5 of MIT OCW Intro to Linear Algebra, and I was really hoping that seeing how determinants are derived from first principles would give me...

Homework Statement Shown In the picture. I went to the prof for help he said and i quote :" don't use laplas expansion to find the determinate, it will take you for ever." Homework Equations I don't even know how to do this. prof had no notes on this and Boas is a god awful book for learning...

Hello! (Wave) Suppose that we are given this $(N+1) \times (N+1)$ matrix: $\begin{bmatrix} -(1+h+\frac{h^2}{2}q(x_0)) & 1 & 0 & 0 & \dots & \dots & 0 \\ -1 & 2+h^2q(x_1) & -1 & 0 & \dots& \dots & 0\\ 0 & -1 & 2+h^2q(x_2) & -1 & 0 & \dots & 0\\ & & & & & & \\ & & & & & & \\ & & & & & & \\...

I just finished writing a computer program that takes as input a number of matrices and computes the inverse of the product of matrices. To test this program, I wanted to input a 3x2 matrix followed by a 2x3 matrix so that the product would be a 3x3 matrix. No matter how hard I try, the...

Homework Statement Show that the determinant of a ##2 \times 2 ## matrix ## \vec\sigma \cdot \vec a ## is invariant under ## \vec \sigma\cdot \vec a \rightarrow \vec \sigma\cdot \vec a' \equiv \exp(\frac{i\vec \sigma \cdot \hat n \phi}{2})\vec \sigma\cdot \vec a \exp(\frac{-i\vec \sigma \cdot...

What is the main idea behind the determinant? What was the main purpose for which it was conceived?

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...

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.

Homework Statement I'm supposed to prove det A = \frac{1}{6} \epsilon_{ijk} \epsilon_{pqr} A_{ip} A_{jq} A_{kr} using the triple scalar product. Homework Equations \frac{1}{6} \epsilon_{ijk} \epsilon_{pqr} A_{ip} A_{jq} A_{ kr} (\vec u \times \vec v) \cdot \vec w = u_i v_j w_k...

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...