Inversion is the process of creating the opposite. Familiar examples include multiplicative inverse $2 \mapsto 1/2$, inverting functions $f(x) \mapsto f^{-1}(x)$, matrix inverse $M \mapsto M^{-1}$ etc. Please include an additional subject tag such as (linear-algebra) or (arithmetic) to help clarify ...

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Properties of frame operator of a matrix

We know many properties of gram matrix $\mathbf{X^TX}$, but what are the properties a frame operator of a matrix i.e., $\mathbf{XX^T}$ and what it tells us about $\mathbf{X}$ ? By properties i mean ...
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1answer
12 views

Expand matrix identity?

What are the intermediate steps to show the following? $$ (I+P)^{−1}(I+P−P) = I−(I+P)^{−1}P $$ I'm looking at the lecture slides here: http://www0.cs.ucl.ac.uk/staff/g.ridgway/mil/mil.pdf
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1answer
26 views

Find $g'(\frac{-1}{2})$ and $g''(\frac{-1}{2})$

Let $f(x)=\frac{x^3}{x^2+1}$, and $g(x)$ is the inverse function of $f(x)$. Then $f(-1)=\frac{-1}{2}$ and $g(\frac{-1}{2})=-1$. Find $g'(\frac{-1}{2})$ and $g''(\frac{-1}{2})$. I have found ...
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61 views

Finding right inverse matrix

Given a $3\times 4$ matrix $A$ such as $$ \begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ \end{pmatrix} ...
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40 views

Proving whether the following are groups or not.

In each case, I am asked to decide whether the indicated pair is a group or not. If so, prove it; if not, show which group axiom fails. (a) $(\dfrac{1}{2}\mathbb{Z}, +)$ where $\dfrac{1}{2} ...
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2answers
42 views

How to show that $\sin^{-1}(x)$ is an increasing function?

I would like to show that $\sin^{-1}$ is an increasing function. In other words, I want to show that $$x_{1} \leq x_{2} \rightarrow \sin^{-1}(x_{1}) \leq \sin^{-1}(x_{2}).$$ I fail to see how to ...
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1answer
26 views

How to Find the Inverse of the Function 13cos(12x)+1

If the domain is between [0,pi/12], how would I get this answer? So far, I have tried the following: 1) Switch 'x' with 'y', so, x=13cos(12y)+1. 2) Try to get 'y' by itself. Therefore, ...
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1answer
34 views

What is $\arccos(\cos(77\pi/4))$?

I did the following steps to get my answer of $-\pi/4$. Subtract as many $2\pi$ as possible from $77\pi/4$, which gives $77\pi/4-18\pi = 5\pi/4$. This leaves the angle in the 3rd quadrant. I believe ...
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1answer
47 views

How to Find the Domain of The Inverse of (4(e^x)-5)/(25(e^x)+12)

These are the steps I have taken so far: In order to find the inverse of the function, I did the following steps: ...
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1answer
523 views

Is there a way to calculate the definite integral of inverse of a 5th degree polynomial?

I want to calculate the definite integral of inverse of a 5th degree polynomial. The problem is that the inverse of the polynomial cannot be calculated (by using Matlab). However without calculating ...
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50 views

Inverse Laplace transform of $\frac{\exp(\frac{\lambda s}{1 - 2s})}{(1 - 2s)^{k/2}}$ (MGF of noncentral chi-squared distribution)

I am trying to use the countour integral to calculate the inverse Laplace transform of the function $$F(s) = \frac{\exp(\frac{\lambda s}{1 - 2s})}{(1 - 2s)^{k/2}} \hspace{1cm}\mathrm{for} \hspace{1cm} ...
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1answer
84 views

Inverse Laplace transform of one complicated function

I want to ask the inverse Laplace transform of the following function: $$F(s) = \frac{1}{s \cdot (1 + a \cdot s)^{m} \cdot (1 + b \cdot s)^{m-k}} \cdot \Bigl[\exp{(\frac{- c \cdot s}{ 1 + b \cdot s } ...
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1answer
17 views

Calculate the inverse of a multi-variable function

How would I calculate the inverse of $F(x,y)=\left( \textrm{arctan} \left(\frac{ay}{x}\right),\frac{x^2+a^2y^2}{2a}\right)$ $a$ is a constant.
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2answers
41 views

If $X$ is a $n$ by $k$ real matrix, and we know that $X^{T}X$ is invertible, is $X$ invertible as well?

If $X$ is a $n$ by $k$ real matrix, and we know that $X^{T}X$ is invertible, we know that $rank(X^{T}X) = n$ by $n$. Can we say that $X$ is invertible as well and hence has rank $n$? thanks!
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34 views

Find Inverse of Matrix A [closed]

If anyone would be able to help me finding the inverse of this matrix $A$ and explaining a little, it would be great! $$A=\begin{bmatrix} -6e^{5t}\sin(3t) & -6e^{4t}\cos(3t)\\ -3e^{5t}\cos(3t) ...
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1answer
23 views

Effective way to calculate the inverse (A+kB)^-1 with k changing and A, B fixed

I have a Simulink modell where I need to calculate $(A+c_k B)^{-1}$ in every time step with $c_k$ changing each iteration. Does someone know any more effective way to do it, instead of calculating a ...
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1answer
48 views

Derivative of Inverse Function from AP

An AP question asks: The function f(x) = x^5 + 3x - 2 passes through the point (1,2). Let f^-1 denote the inverse of f. Then (f^-1)(2) equals? The inverse of this function should be y^5 + 3y - 2. ...
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2answers
80 views

Inverse Laplace Transfrom of $s^{-1}e^{-a\sqrt{s} + b/s}$

I am trying to find the inverse Laplace transform for following function and it seems almost impossible for me to find the answer. Can anyone help me please with final answer and also the way to get ...
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4answers
36 views

Find the interval for which $2\arctan x + \arcsin \frac{2x}{1+x^{2}}$ is an independent of x?

I used formula and simplified the expression to $\Rightarrow 4\tan^{-1} x$
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2answers
39 views

Show that function $f(x)=x^3 - 3x^2 +3x +4$ is invertible and Calculate the inverse [closed]

$f(x)=x^3 - 3x^2 +3x +4$ is a full cubic function. I do not have any clue. If somehow i could factor it... Online solvers only give solution no explaination.
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1answer
58 views

How to find the inverse of an upper triangular matrix

I want to find the inverse of an upper triangular matrix in an efficient way. I googled a lot, but all the articles discussed about a lower triangular matrix. Is it possible to edit the matlab code ...
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1answer
31 views

Calculation of inverse function

My question is physics related but the problem itself is just math. I have an expression for a refractive index depending on the wave length $\lambda$: $$ ...
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1answer
44 views

Inverse of $x\log(x)$ for $x>1$

Let $y(x)=x\log(x)$ for $x>1$. Is it possible to write down the inverse function explicitly? Has this inverse function been named? (For example, the Bessel functions are "named" but cannot be ...
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81 views

Inverse of a matrix!

Let $I_n$ be the $n$ by $n$ identity matrix and $b$ and $c$ be two vectors in ${\mathbb R}^n$ such that $b^Tc\ne 0$. Then one can easily see that the $n+1$ by $n+1$ matrix $X$ defined as following $$ ...
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2answers
40 views

Derivative of the inverse of a matrix

I've seen in a scientific paper this equation: $\frac{\delta K^{-1}}{\delta p} = -K^{-1}\frac{\delta K}{\delta p}K^{-1}$ where K is a $n\times n$ matrix which depends on $p$. In my calculations I ...
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2answers
61 views

If $y=ax^be^{-cx}$ then $x=g(y)$, find $g$

I have this function $$y=0.384394\cdot x^{0.341429}\cdot e^{-0.004749 x}$$ Based on this function I would like to know how I can I get $x=g(y)$.
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1answer
26 views

Inverting functions containing ceiling

I have a recurrence relation for a countably infinite sequence that contains the integers divisible by 5 but not by 7. The relation I came up with is: $5((n-1) + \lceil \frac{n}{6} \rceil)$ The ...
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31 views

Order of the sum of elements of the inverse of a matrix

For each $T$, let $A_T$ be a $T\times T$ matrix of real numbers. let $e_T$ be the $T\times 1$ vector of ones. Assume that the sum of all entries of the matrix $A_T$ divided by $T^2$ is limited as $T$ ...
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26 views

Differential Equation for Inverse of a Function given the ODE for the function

Assume we have and ODE for a function $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ in the form of \begin{equation} \mathbf{J}f \times F(x)=D(\lambda_i)\times f(x) \end{equation} where $\mathbf{J}$ denotes ...
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1answer
53 views

How to estimate the size of the neighborhoods in the Inverse Function Theorem

Given a function $f:U \subset V\to W$ such that $\textbf{D}f(x_0)\neq 0$ for some $x_0$. How to estimate the neighborhood for which it's invertible? Assuming the second derivative exists and is ...
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1answer
137 views

Which graphs do have invertible adjacency matrices?

I would like to know if there is any class of graphs for which the adjacency matrices are invertible. At this moment I am aware of only the class of graphs $n K_2$ which is the disjoint union of $n$ ...
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19 views

Inverse function of a function of Laplacian Matrix

I have a function $f:\mathbb R^n \rightarrow \mathbb R^n$ defined by $f(\mathbf{X})=L(\mathbf{X})\mathbf{X}$ where $L(\mathbf{X})$ is a (nonlinear) Laplacian matrix of an undirected but of-course ...
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2answers
73 views

Find a formula relating $\operatorname{arcsin}(x)$ and $\operatorname{arccos}⁡(x)$

From the formula $\sin\left(\frac{π}{2}−x\right)=\cos x$, find a formula relating $\operatorname{arcsin}(x)$ and $\operatorname{arccos}⁡(x)$. I have figured out that the domain of $x$ is ...
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1answer
67 views

Inverse of a function's integral

The function $g$ is strictly positive. Let the function $f$ be defined as $$f(x) = \int_0^x g(u) du$$ Is there a way to express $f^{-1}(x)$ in terms of $g$?
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74 views

Help with Inverse Function and Composition of Functions?

I'm currently doing work on discrete mathematics in my free time and am having some difficulties with understanding some questions pertaining to Relations and Functions. To be specific, I'm stuck on ...
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75 views

Find the inverse of the $n\times n$ matrix whose entries are given by $a_{ij} = \max (i,j)$

The actual question on the past papers is "Let $n\ge 1$ be an integer and consider the $n\times n$ matrix $A$ whose entries are given by $a_{ij} = \max(i,j)$ for all $1\le i,j\le n$. Show that $A$ is ...
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177 views

Show that f(x)=e^x from set of reals to set of reals is not invertible…

Yes, this is my question... How can you prove this? That $f(x)=e^x$ from the set of reals to the set of reals is not invertible, but if the codomain is restricted to the set of positive real numbers, ...
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5answers
108 views

Prove that if $f$ is increasing then so is $f^{-1}$

Prove that if $f$ is increasing then so is $f^{-1}$, when $f$ is a one-to-one function. I'm having trouble figuring out how to get started with this question. I'm assuming it has something to do with ...
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1answer
26 views

Find inverse $z$-transform of $\frac{5}{z^{2}-z-6}$

How can I find inverse z transform of $$X(z)=\frac{5}{z^{2}-z-6}$$ What I did: first i factored denominator and i got (z+2)(z-3), now we get A(-2^{n}) + b(3^{n}). To get A and B i used Partial ...
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1answer
38 views

Cofactor expansion to check if matrices is invertible.

I gave question regarding a co-factor expansion question. I understand that an easy way to check if a matrices is invertible is to do co-factor expansion and if $A \ne 0$ then its invertible. I'm ...
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5 views

Inverse Theory: Setting up data kernal G for distance between two points problem

I'm looking to put the simple equation $$d_i = \frac{\sqrt[]{(x_1-x_0) ^2 + (y_1 - y_0)^2 + (z_1 - z_0)^2 }} {v}$$ into the inverse form: $$ d = Gm $$ where $$ d_i = [t_1 , t_2 , . . . , t_i ]^T ...
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1answer
42 views

Find inverse $z$-transform of $\dfrac{(z-1)^2}{z^3}$

How can I find inverse z transform of $$X(z)=\frac{(z-1)^{2}}{z^{3}}$$ What I did: I am thinking to do Partial Fraction Decomposition or long division. Is there another method ?
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1answer
43 views

Invertibility of symmetric matrix [closed]

Is a symmetric matrix always invertible?
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1answer
22 views

Write $(h \circ f)^{-1}$ when $h(x)= x \ln(3 + x)$ and $f(x) = x^2 − x$

I have arrived up to a point but haven't solved it yet: $$(h \circ f)^{-1} = y= (x ^2 − x )· \ln(3 + x^ 2 − x)$$ $$ x = (y^ 2 − y )\cdot \ln(3 + y^ 2 − y)$$ Any suggestions? Thank you
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1answer
36 views

Inverse Matrices. Unsure how to solve it.

Find the inverse of matrix $A$. Can some please show me how to do this question.I've been attempting this question for quite awhile now, although don't know how to proceed. $$A=\begin{bmatrix} ...
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1answer
30 views

Inverse of a Function with Complex Variables

I'm taking Abstract Algebra, and we're currently covering isometries of the Real and Complex plane. I'm going through and studying for our first midterm, and I'm working on a problem that asks to show ...
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48 views

Inverse function theorem - good proof

I am looking for a reference which give a full demonstration of the inverse function theorem (let say in Banach spaces) where we can have estimates of the bounds of the neighbourhoods that we build to ...
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20 views

Proof of the surjectivity of $f$ given it has a right inverse gives $f$ not-a-function?

I am working on a proof of the surjectivity of $f:X\rightarrow Y$ given that it has a right iverse $g_R:Y\rightarrow X$ such that $f(g_R(y))=y\,\,\, \forall y\in Y$. My question stems from this. We ...
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2answers
41 views

Given $f(x)$ with inverse $g$, find $g'(2)$

Problem Given the function $$f(x) = \ln^3(x) - 2\ln^2(x) + \ln(x)$$ defined for $$x\in[e, e^3]$$ show that the function has an inverse $g$ on the given interval, and find $g'(2)$ Progress I have ...
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2answers
106 views

Find a formula relating $\arcsin (x)$ and $\arccos (x)$ [duplicate]

From the formula $\sin(\frac{\pi}{2}-x)=\cos x$, find a formula relating $\arcsin (x)$ and $\arccos (x)$. I have no idea where to start.