Inverse Definition and 1000 Threads

$\textsf{Find the inverse of matrix} $ $$A=\left| \begin{array}{rrr} 1&0&2\\ 1&0&0 \\ 3&2&0 \end{array} \right|$$ $\textsf{by method of adjoint matrix }\\$ $\textsf{adj $A = |C_{ij}|^T$}\\$ $\textsf{det $A =4$}\\$ $\textsf{so then}\\$ $A^{-1}=\frac{1}{det A}(adj A)= \frac{1}{4}(adj A)$...

Defining dS2 as gijdxidxj and given dS2 = (dx1)2 + (dx2)2 + 4(dx1)(dx2). Find gijNow here is my take on the solution: Since the cross terms are present in the line element equation, we can say that it's a non-orthogonal system. So what I did was express the metric tensor in form of a 2x2...

I am working through Lessons in Particle Physics by Luis Anchordoqui and Francis Halzen. The link is https://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1271v2.pdf. I am on page 21. Between equations (1.5.53) and (1.5.54), the authors make the following statement: ##S^\dagger ( \Lambda ) = \gamma ^0...

Homework Statement why this formula works ? Homework EquationsThe Attempt at a Solution when i take the derivative of the right side ,,, there is an additional "a" in the numerator in place of 1,, why the derivative of arcsine of (u/a) not exactly same with the expression under the integral sign

Let's say you can bend a paper...how about bending it upward. a slope I'm saying as we saw spactime in 3d...we all know how it looks..the lines are attracted toward Earth but why doesn't it deflects them and maybe negative mass is linked with it. In other words, someone under the trampoline...

Hi all, so I'm not sure if what I'm asking is trivial or interesting, but is there any general or canonical way to interpret say, The follwing operator? (Specifically in the study of quantum mechanics): A = 1/(d/dx) (I do not mean d-1/dx-1, which is the antiderivative operator ) How would...

Homework Statement In the inverse Compton scattering there is a particle, with energy ##E## in the laboratory frame and mass at rest ##m##, which collide head on with a photon with energy ##E_\gamma##. Finding the maximum energy the photon can have after being scattered. The Attempt at a...

Homework Statement Consider the function ##f\left(x\right)=\sqrt {x+2}##. Determine if the function is a one-to-one function, If so, find ##f^{-1}\left(x\right)## and state the domain and range of ##f\left(x\right)## and ##f^{-1}\left(x\right)## Homework Equations N/A The Attempt at a...

Homework Statement I am given this equation: and asked to solve using Laplace transforms The Attempt at a Solution This is what I did: This seemed logical to me, I used partial fractions and it stayed pretty simple. This is what the solutions my prof posted do: Is my answer equivalent...

Hey! :o Let A be a regular ($n\times n$)-Matrix, for which the Gauss algorithm is possible. If we choose as the right side $b$ the unit vectors $$e^{(1)}=(1, 0, \ldots , 0)^T, \ldots , e^{(n)}=(0, \ldots , 0, 1 )^T$$ and calculate the corresponding solutions $x^{(1)}, \ldots , x^{(n)}$ then...

Hi PF! I'm reviewing a text and the author writes where ##g## is an arbitrary function and ##B## is a differential operator. ##Bo## is a parameter. Then the author states the inverse of ##B## is where ##G## is the Green's function of ##B##. Can someone explain how we know this?

The following matrix A is, \begin{equation} A= \begin{bmatrix} a+b-\sigma\cdot p & -x_1 \\ x_2 & a-b-\sigma\cdot p \end{bmatrix} \end{equation} The inversion of matrix A is, \begin{equation} A^{-1}= \frac{\begin{bmatrix} a-b-\sigma\cdot p & x_1 \\ -x_2 & a+b-\sigma\cdot p...

Homework Statement Find the trace of a ##4\times 4## matrix ##\mathbb U=exp(\mathbb A)##, where $$\mathbb A = \begin {pmatrix} 0&0&0&{\frac {\pi}{4}}\\ 0&0&{\frac {\pi}{4}}&0\\ 0&{\frac {\pi}{4}}&0&0\\ {\frac {\pi}{4}}&0&0&0 \end {pmatrix}$$ Homework Equations $$e^{(\mathbb A)}=\mathbb P...

find the value of $$df^{-1}/dx at $x=f(a)$$ $$f(x)=x^3-6x^2-3$$ $$x \ge 4$$ $$a=3$$ ok the inverse would be $$x=y^3-6y^2-3$$ but don't see how to isolate $y$ or if we need to

Homework Statement For example : How to inverse z-domain function (z2+3z+7)/(z2+4z+3) The Attempt at a Solution Whatever I use partial fraction to simply the z-domain function, I cannot continue the next step, such as 1/(z+3)

Hi, I'm struggling to understand something. Does domain restriction work the same way for composition of inverse functions as it does for other composite functions? I would assume it does, but the end result seems counter-intuitive. For example: If I have the function f(x) = 1/(1+x), with...

How do I solve this. Getting an inverse alone seems to be a quite long route. Is there an easier way of doing this?

Find the inverse of f(x) = 2/(x - 3). Let y = f(x) y = 2/(x - 3) Replace y for x. x = 2/(y - 3) x(y - 3) = 2 Solve for y. xy - 3x = 2 xy = 2 + 3x y = (2 + 3x)/x Replace y with f^-1 (x). f^-1(x) = (2 + 3x)/x 1. Is f^-1(x) the inverse of f(x)? 2. What does f(x) and f^-1(x) look like...

Homework Statement I'm given a force law is F = \frac{-k}{r^3} and that initially, the particle is in a circular orbit the particle is given an impulse parallel and in the opposite direction to its velocity find the distance from the center for the particle as a function of time. Homework...

Homework Statement Prove that [adj(A)]^{-1} = adj(A^{-1}) Homework EquationsThe Attempt at a Solution Ok. So if 1/det(a) * adj(a) = A^{-1} is true, then adj(A) = A^{-1} det(A) then [adj(A)]^{-1} = 1/det(A^{-1}) * adj(A^{-1}) * det(A) now the statement would be proved if det(A)...

Homework Statement I was hoping someone could just verify this solution is accurate. p(x) = 0 , x < 0 4x, x < .5 -4x + 4 , .5 <= x < 1 Find CDF and Inverse of the CDF. Homework EquationsThe Attempt at a Solution CDF = 0 , x < 0 2x^2 ...

Hello all, I have posted on Physics Forums a few times in the past, but mostly for help with my old physics classes and not anything in the real world. Part of my work involves radiography, but it is generally done in a field environment where we just shut down large sections of land to safely...

If the inverse square law for gravity varies with distance or distribution of matter, might the need for “dark matter” be obviated?

2 sound waves that are mathematical polarities cancel each other out according to my audio engineering book. I thought energy cannot be destroyed, just changed. Am I wrong? What happens to the energy? Same question could be applied to matter and anti matter right?

Hi, I'm afraid I not very good at these questions just yet and would like a walk through a bit better than the one I was given by my tutors. Thank you, please refer to the inline image.

Hello buddies! Please, check out these equations... Tell me, please, are they mathematically correct or not? I need a simple YES/NO answer. I have not sufficient knowledge to understand them. I just need to know whether they are correct... Thank you! P.S. Am is amplitude; I guess it is a...

Homework Statement Q/ in this inverse Fourier problem, how did he come with the results of integration of (Sinc) function and how did he come up with those results of integration with the inverse part (as in the attached picture) here is the problem: https://i.imgur.com/Ir3TQIN.png Homework...

I know that one can easily prove the result that a function is injective if and only if that function has a left inverse. But is there intuitive reason for this? Same goes for the fact that having a right inverse is equivalent to being surjective. Why are the properties of injectivity and...

As I understand it, in the context of cosmological perturbation theory, one expands the metric tensor around a background metric (in this case Minkowski spacetime) as $$g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}$$ where ##h_{\mu\nu}## is a metric tensor and ##\kappa <<1##. My question is, how...

We use A = I.A as equation and then by transforming only A of LHS and I of RHS we come to I = P.A and we say that P is the inverse of matrix A My question is that why we only tranform A and I, why A of RHS is left as it is during the transformation, or why transformation do not take place in...

the graph of sin inverse (sin x) between the domain of ( -pi/2,pi/2) is y = x. but after it crosses that domain of course the expression won't be the same anymore because sin inverse has its principle value as ( - pi/2, pi/2) due to sin x many to one natured function. now the way these...

Hello, I want to integrate this expression : ∫ (x5 + ax4 + bx3 + cx2 + dx)-1 between xmin>0 and xmax>0 a is positive but b, c and d can be positive or negative. I have no idea to integrate this expression... Do you have methods to do this ? Thanks in advance !

Homework Statement Find the inverse of ##A = \begin{bmatrix} 1 & \dfrac12 & & \cdots && \dfrac1n \\\dfrac12 & \dfrac13 && \cdots && \dfrac1{n+1} \\ \vdots & \vdots && && \vdots \\ \dfrac1n & \dfrac1{n+1} && \cdots && \dfrac1{2n-1}\end{bmatrix}## Homework EquationsThe Attempt at a SolutionI...

Homework Statement :[/B] Solve for ##x ##: $$ \sin ^{-1} {x} +\sin ^{-1} {(1-x)} =\cos ^{-1} {x} $$ Answer given: ##0## or ##\frac {1}{2}##. Homework Equations :[/B] All relevant formulae on inverse circular functions may be used. The Attempt at a Solution :[/B] Please see the pic below...

Homework Statement I got the laplace transfer function H(s) = 1/(s + 2) and I'm suppose to find the inverse Z transform by first converting to H(z) by s = Ts/2*(z-1)/(z+1) Then do inverse Z-transform using the "displacement rule" - Never heard of. Homework Equations H(s) = 1/(s + 2) s =...

Homework Statement Using the following information, find the matrix A (I+2A)-1 = [-1 2] [4 5] Homework Equations AA-1 = I The Attempt at a Solution none. I have no idea how should I start. The inverse on the whole left side is driving me crazy.

Homework Statement Using magnetic field over electric field Homework Equations no equation needed The Attempt at a Solution THis may not make sense but did an experiment dealing with the inverse square law and we measured the magnetic field in this case. Want to know is there some type of...

Hi everyone, I am currently working on a subject that involves a lot of 4th order tensors computations including double dot product and inverse of fourth order tensors. First the definitions so that we are on the same page. What I call the double dot product is : $$ (A:B)_{ijkl} =...

Then it goes on explaining how Gauss law would fail because for a very large surface, E field would be vanish with flux through it and though we can calculate div for this field it won't depend on source density. But I don't get what makes this particular function so evil that it would break...

Homework Statement I have to take the inverse Laplace of this function (xoms+bxo)/(ms2+bs+k) this can not be broken into partial fractions because it just gives me the same thing I started with. How is this done? This is coming from the laplace of the position function for a harmonic oscillator...

Isn't Inverse Compton scattering just the Doppler's effect? A fast moving electron gets slowed down by a photon. This photon then becomes blue shifted, becoming a gamma ray. Kinda makes sense.

Homework Statement F(t) = sqrt(π/2)e-t for t>0 F(t) = sqrt(π/2)et for t<0 In other words the question asks to solve this integral: 1/sqrt(2π) ∫F(t)eitxdt and show that it equals 1/(1+x2) Homework Equations F(t) = sqrt(π/2)e-t for t>0 F(t) = sqrt(π/2)et for t<0 1/sqrt(2π) ∫F(t)eitxdt The Attempt...

Homework Statement I want to find the Fourier series of the sawtooth function in terms of real sine and cosine functions by using the formula: $$f_p (t)=\sum^\infty_{k=-\infty} c_k \exp \left(j2\pi \frac{k}{T}t \right) \tag{1}$$ This gives the Fourier series of a periodic function, with the...

Hi, got a question I'm stuck on.. Write down a matrix P which will diagonalise A and write down the corresponding diagonal matrix D, where D = P^-􀀀1AP. You do not have to calculate P^-1 Ive got all the eigenvalues and eigenvectors for A, and thus have the Matrix P, which has a determinant of...

I need help in understanding how Jacobian Elliptic Functions are interpreted as inverses of Elliptic Functions. Please reference the wiki page on Jacobian Elliptic functions: https://en.wikipedia.org/wiki/Jacobi_elliptic_functions For example, if $$u=u(φ,m)$$ is defined as $$u(φ,m) =...

Homework Statement Let ##A## be an ##m \times n## matrix with rank ##m##. Prove that there exists an ##n \times m## matrix ##B## such that ##AB= I_m## Homework EquationsThe Attempt at a Solution So here is how far I get. I am given that ##A## has rank ##m##. Since ##L_A(x) = Ax## is a map...

if y = tan inverse (cot x) + cot inverse (tan x)

if y = sin inverse (x square + 2x) find dy/dx

Let ##\Lambda## be a Lorentz transformation. The matrix representing the Lorentz transformation is written as ##\Lambda^\mu{}_\nu##, the first index referring to the rows and the second index referring to columns. The defining relation (necessary and sufficient) for Lorentz transforms is...

Hi All, Which are the ways one can geometrically obtain, given a line segment AB with length x and an unitary segment OC, a line segment with length 1/x ? Straight edge and compass are allowed (also some auxilliary curve). Best wishes, DaTario