Inverses include: multiplicative inverse of a number (reciprocal), inverse function, matrix inverse, etc. A subject tag such as (linear-algebra), (algebra-precalculus) or (arithmetic) should be added to clarify in which sense "inverse" is used. This tag should never be the only tag on a question.

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Inverses where $f(g(x))=x\neq g(f(x))$

If $f: \mathbb{Z\to Z}$ and $G: \mathbb{Z\to Z}$, find $f$, and $g$ such that $f(g(x))=x\neq g(f(x))$. I can find lots of $f$ and $g$ that aren't equal when composed with each other, but I have no ...
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82 views

Can $A^{T}(AA^{T})^{-1}A$ be simplified?

Let $A$ is an $m\times n$ ($m<n$) real matrix with full positive entries and $\text{Rank}(A)=m$. Thus $(AA^{T})^{-1}$ is an $m\times m$ symmetric $M$-matrix since $AA^{T}$ is nonnegtive and ...
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55 views

Show that $Y$ is invertible

Let X be a $40\times40$ matrix such that $X^3 = 2I$. I want to show that $Y= X^2 -2X + 2I$ is invertible as well. I tried working with the equations to see if I can get Y as a product of matrices ...
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179 views

Given a finite metric space, are the matrices of triangle inequality errors invertible?

I have been working on some problems regarding finite metric spaces and have already proven/positively answered the following statement/question if the underlying metric has additional properties. Now ...
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210 views

Statement about $(I-A)^{-1}$ matrices

Let $A \in \mathbb{R}^{n \times n}$ and let denote $I$ the $n \times n$ identitiy matrix. Theorem. If $(I-A)$ is invertible and $(I-A)^{-1}$ is a nonnegative matrix and there is such a diagonal ...
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169 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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97 views

Inverse of $\frac{1-e^{-x}}{x}$ on $(0,1)$

I am trying to invert (or to estimate the inverse of) $$y=\frac{1-e^{-x}}{x}$$ for $y\in(0,1)$. The function 'looks' monotonically decreasing between $x=0$ and $x=\infty$, but I have not been able to ...
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Inverse of a Function exists iff Function is bijective

How to mathematically prove that inverse of a function, let's say, $f^{-1}$, exists, if and only if $f$ is bijective? I know how to prove it using diagrams but I'm looking for a rather mathematical ...
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1answer
58 views

Finding the inverse of a map from $CP^1$ to $S^2$

Given the map: $$f:CP^1 \to S^2\ ,\ f[z:w] = \left(\frac{2\mbox{Re}(w\bar{z})}{|w|^2+|z|^2},\frac{2\mbox{Im}(w\bar{z})}{|w|^2+|z|^2}, \frac{|w|^2-|z|^2}{|w|^2+|z|^2}\right)$$ How would I go about ...
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601 views

Super logarithmic inverse of tetration

What's the super logarithmic inverse of tetration for $\bf{^{2}{x}}$? Is it $slog^{x}_{2}$?
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Find Functions That Can Be Inverted from Their Sums

I have the following situation:$$ f_1(x_1) + f_1(x_2) + f_1(x_3) + \cdots + f_1(x_n) = c_1\\ f_2(x_1) + f_2(x_2) + f_2(x_3) + \cdots + f_2(x_n) = c_2\\ \vdots\\ f_n(x_1) + f_n(x_2) + f_n(x_3) + \...
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56 views

Using the Inverse Function Theorem prove that $(\sin^{-1}x)'$ = $\frac{1}{\sqrt{1-x^2}}$.

Using the Inverse Function Theorem prove that $(\sin^{-1}x)'$ = $\frac{1}{\sqrt{1-x^2}}$. Proof: Let $f(x) = \sin x$, for $x$ in $(-1,1)$. Then let $x_{0}$ be in (-1,1). Then $f'(x_{0})$ = $\cos(x_{...
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368 views

Find inverse for the closed-form expression of linear recurrence relation

I am trying to find an inverse of the following formula: $$ a_n=\frac{2+\sqrt{6}}{4}(1+\sqrt{6})^n+\frac{2-\sqrt{6}}{4}(1-\sqrt{6})^n $$ This formula is a closed-form expression of a linear ...
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1answer
745 views

Inverse of elliptic integral of second kind

The Wikipedia articles on elliptic integral and elliptic functions state that “elliptic functions were discovered as inverse functions of elliptic integrals.” Some elliptic functions have names and ...
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Eigenvectors of difference of inverse matrices

I have two matrices $A$ and $B$, symmetric and positive semi-definite (in fact, they are covariance matrices), and I am interested in computing the eigenvectors of the matrix $A^{-1}-B^{-1}$. From ...
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208 views

How to find inverse of $\sin(x) + \sin(2x) = y$?

I was wondering if there were any way to solve the equation $$\sin(x) + \sin(2x) = y$$ in terms of $x$. This of course would allow us to express the inverse for this function on $-\frac{\pi}{4}$ to $\...
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90 views

Is the inverse of any elementary function asymptotic to some elementary function?

Is the functional inverse of any elementary function asymptotic to some elementary function ? For instance Lambert's $W(z)$ is asymptotic to $ln(z)$. See http://mathworld.wolfram.com/LambertW-Function....
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Explicit quasi-inverse of Künneth-isomorphism?

With $A_X$ the complex of $\mathbb{R}$-differential forms on $X$, the Künneth theorem states that \begin{align*} A_X \otimes A_Y &\to A_{X \times Y}, \\ (\omega,\eta) &\mapsto {\rm pr}_X^\...
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Is $a^{-1} + b^{-1} = (a + b)^{-1}$ always true for Abelian group?

I get the equation $a^{-1} + b^{-1} = (a + b)^{-1}$ from ordinary + operation. For ordinary + operation I mean $a^{-1} = -a$. It is also true for * of rational numbers $3^{-1}*4^{-1} = \frac{1}{3} * \...
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Is there a difference between $(x)^{\frac{1}{n}} $ and $\sqrt[n]{x}$?

Is there a difference between $(x)^{\frac{1}{n}}$ and $ \sqrt[n]{x}$ ? I'm confused with this topic. Any ideas or examples ? If $(x)^{\frac{1}{n}} = \sqrt[n]{x}$ Consider $x=\frac{-b \pm \sqrt{b^2-...
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Will inverse functions, and functions always meet at the line $y=X$?

If I have a function, the inverse function, by definition will be a reflection of the original function in the line $y=X$, so if I wanted to find the point of intersection, instead of solving it with ...
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how to find inverse of a matrix in $\Bbb Z_5$

how to find inverse of a matrix in $\Bbb Z_5$ please help me explicitly how to find the inverse of matrix below, what I was thinking that to find inverses separately of the each term in $\Bbb Z_5$ and ...
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135 views

Finding the inverse of $f(x) = x^3 + x$

How can one find the inverse of functions like $f(x) = x^3 + x$? I know how to do it for explicit quadratic functions; how do I express $x$ as a function of $y$ here?
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Definition of Inverse in Linear and Abstract Algebra

In a linear algebra text, the following is the definition of the inverse of a matrix An $n\times n$ matrix $A$ is invertible when there exists an $n \times n$ matrix $B$ such that $$AB = BA = I_n$$...
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Finding inverse of a function $h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$

I have a function: $$h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$$ With just pen and paper, how can I determine if there exists an inverse function? Am I supposed to sketch it on paper to see if it can ...
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How should I understand $f^{-1}(E):=\{x\in A:f(x)\in E\}$?

I understand the concept, but I still can't figure out how to read the notation: $$f^{-1}(E):=\{x\in A:f(x)\in E\}$$ I understood the concept due to the examples, not with the notation. Can someone ...
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Inverse of the sum of the inverse of two matrices

I need to compute $ (A^{-1} + B^{-1})^{-1} $. Both $A$ and $B$ are symmetric and $A$ is invertible and PSD. I already know $B^{-1}$ and $A$, but I don't have $A^{-1}$ and $B$. Is there a formula to ...
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97 views

Prove that if $AA^T=A$ then $A^3=A$

The approach I'd like to use to prove this particular property necessitates that $A$ be invertible, but I don't wish to assume this (though it would certainly make the task simpler). Is there some ...
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641 views

“Orthogonal” Rectangular Matrix

Is it possible to have a matrix $\mathbf B \in \mathbb R^{m\times n}$ such that it satisfies: $$\mathbf B^T\cdot\mathbf B = \mathbf I_n$$ Where $\mathbf I_n$ is the $n\times n$ identity matrix. Or ...
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$f \circ g =\operatorname{ id}$ and $g \circ f \neq \operatorname{id}$?

Are there two functions $f$ and $g$ s.t. $$f \circ g = \operatorname{id}$$ but $$g \circ f \neq \operatorname{id}?$$ Could someone give an example or a proof that this is impossible? This must be ...
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How to find inverse of the function $f(x)=\sin(x)\ln(x)$

My friend asked me to solve it, but I can't. If $f(x)=\sin(x)\ln(x)$, what is $f^{-1}(x)$? I have no idea how to find the solution. I try to find $$\frac{dx}{dy}=\frac{1}{\frac{\sin(x)}{x}+\ln(x)\...
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How to invert this function? (Inverse exponential function with arctan)

How to invert this function? $$ y = e^{\arctan(x^5)} $$
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The relation between an exponential function and a logarithmic function

I have been told multiple times that the logarithmic function is the inverse of the exponential function and vice versa. My question is; what are the implications of this? How can we see that they're ...
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An ill-conditioned matrix

If C is an ill-conditioned matrix and I want to get the inverse, one way is to take a pseudo-inverse of some sort. Instead, is the following, which uses the (normal) inverse, also a way to deal with ...
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When A and B are of different order given the $\det(AB)$,then calculate $\det(BA)$

Let 'A' be a $2 \times 3$ matrix where as B be a $3 \times 2$ matrix if $\det(AB) = 4$ the find value of the $\det(BA)$ My attempt: I took A = $$ \begin{bmatrix} 2 & 0 &0\\ ...
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155 views

Non-associative: set with a binary operation, but has inverses and identity

I've been thinking about an example of some set with a binary operation which would satisfy all axioms of groups except for associativity. I'm new to Group Theory, so I would appreciate your ...
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206 views

Calculate inverse of arbitrarily sized, lower triangle matrix with a specific pattern.

I have a matrix of the following form: $$A=\begin{bmatrix} 2 & 0 & 0 & 0 \\-1 & 2 & 0 & 0\\ 0 & -1 & 2 & 0\\ 0 & 0 & -1 & 2 \end{bmatrix}$$ which, in ...
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Taylor series of the inverse of $x^4+x$

I would like to expand the inverse function of $$g(x) := x^4+x $$ in a taylor series at the point x = 0. I calculated the first and second derivate at x = 0 with the rule of the derivation of an ...
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mean and variance of reciprocal normal distribution

If $X$ is a normal distributed with mean $\mu$ and variance $\sigma^2$. What would be the mean and variance of $Y = \dfrac{1}{X}$
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Finding inverse of a matrix

This question is in my assignment. We are not allowed to use any symbol to represent any elementary row and column operations used in the solution. We must solve it step-by-step. Please help me to ...
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Matrix Inverses

So in class we have been discussing matrix inverses and the quickest way that I know of is to get a matrix A, and put it side by side with the identity matrix, like $[A|I_{n}]$ and apply the Gauss-...
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335 views

Inverse function notation

Suppose $f$ and $g$ are functions that fail to be one-to-one, but $f+g$ is one-to-one. Has anyone ever seen the notation $(f+g)^{-1}$ for the inverse function in that situation? (I find myself ...
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Tool for expressing $x=f^{-1}(y)$ if $y=f(x)$ is given

I have many equations of nature - $y=ax^{12}+bx^5+5x^4+1$ For all these equations, I need to express x in terms of y. What tool or software would you recommend for this? I would much prefer to ...
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63 views

Looking for a commutative ring satisfying certain conditions

I'm looking for a commutative ring $R$ (with unit) which is of characteristic 2 and which possesses elements $x$ and $y$ such that the following holds $x^2$ and $y^2$ are inverses of one another but $...
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158 views

Inverse of $f(x)=\sin(x)+x$

What is the inverse of $$f(x)=\sin(x)+x.$$ I thought about it for a while but I couldn't figure it out and I couldn't find the answer on the internet. What about $$f(x)=\sin(a \cdot x)+x$$ where ...
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If matrix A is invertible, is it diagonalizable as well?

If a matrix A is invertible, then it is diagonalizable. Is it true or false?
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462 views

Prove the inverse of the Hilbert matrix has integer entries [duplicate]

$1 \frac{1}{2} ... \frac{1}{n}$ $\frac{1}{2} \frac{1}{3} ... \frac{1}{n+1}$ $.$ $.$ $.$ $\frac{1}{n} \frac{1}{n+1} ... \frac{1}{2n-1}$ Does the inverse of this matrix ...
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Solve equation $\tfrac 1x (e^x-1) = \alpha$

I have the equation $\tfrac 1x (e^x-1) = \alpha$ for an positive $\alpha \in \mathbb{R}^+$ which I want to solve for $x\in \mathbb R$ (most of all I am interested in the solution $x > 0$ for $\...
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To invert a Matrix, Condition number should be less than what?

I see that there is a matlab tag in this site, so I ask my question here and not in stackoverflow although it is also related to programming in matlab. I am going to invert a positive definite matrix ...
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555 views

Invertible Derivative

I'm trying to brush up on some differential geometry, but there's a subtle point I don't understand. Suppose $h$ is a diffeomorphism. Then the lecture notes here suggest that it's derivative $df_x$ is ...