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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Groupoid element with multiple inverse elements?

Simply put, is there a groupoid whose element can have multiple inverse elements? I know how to prove that elements of a semigroup have unique inverses, but this is a bit diferent... If there is such ...
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33 views

Uniqueness of right identity

I'm working on a problem which says the existence of a unique right identity and left inverse (which may not be unique) on a set with binary operation constructs a group. Of course I know that the set ...
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34 views

Show the inverse of the One to One function

Is anyone able to guide me in the right direction for this question. This is for a beginner assembly language class. This is an online course so I am unable to ask the professor for guidance. Show ...
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1answer
15 views

Finding selective inverses for functions: how?

So, I have a function $f(x) := \frac{3x}{7+x^2}$ and the question is that the domain is defined such as $x \geq a$, find the minimum value for $a$ for which there exists an inverse of the function. ...
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96 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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1answer
28 views

Find injectivity/ prove one to one algebraically of $ \frac{|x|x}{\sqrt{x^4-81}}$

I'm trying to find the range of this function: $$ \frac{|x|x}{\sqrt{x^4-81}}. $$ However to find the range I have to first prove one to one, then find domain of inverse. I can't prove injectivity ...
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2answers
100 views

Show $\int_0^a f(x)\,dx + \int_0^b f^{-1}(x)\,dx \ge ab$ for strictly increasing function $f(x)$

Let $f: [0, \infty) \to [0, \infty)$ be continuous and strictly increasing and with $f(0) = 0$. Prove that $$\int_0^a f(x)\,dx + \int_0^b f^{-1}(x)\,dx \ge ab$$ for any $a, b > 0$, and give a ...
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29 views

Does the arccos itself contain all solutions or just one solution?

For the equation $$\cos(x)=\frac{1}{2}$$ All solutions are: $$x=\pm\frac{\pi}{3}+p2\pi,\quad p\in\mathbb Z\:.$$ To find these solutions, I use the inverse cosine ($\arccos$ or $\cos^{-1}$). Is the ...
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3answers
90 views

Regarding an inverse trigonometric equation.

I tried to find the solutions of this equation $$ \arctan\left(\frac{2x}{1-x^2}\right)+\text{arccot}\left(\frac{1-x^2}{2x}\right)=\frac{2\pi}{3} $$ I got solutions $\frac{1}{\sqrt{3}}$ and $-\sqrt3$. ...
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1answer
28 views

f differentiable map of finite dimensional vector spaces, with derivative injective. Why is f injective?

Suppose A and B are finite dimensional vector spaces, $U\subseteq{A}$ is an open subset, $a\in U$ and $f:U\rightarrow B$ is $C^\infty$ with $(Df)_a$ injective. I need helping showing that there exists ...
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41 views

Relation between $x,y,z$…Exponent problem…

The given equation is- $\sqrt[x]{75} = \sqrt[y]{45} =\sqrt[z]{15}$ Now,it is required to prove $x+y=3z$. I want the simplest possible solution.Thanks in advance.
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4answers
651 views

is it true every left inverse of a matrix is also right inverse of it?

I am wondering that, consider there are $m$ linear equations with $n$ unknowns. We can represent it as $AX=B$. Let $L$ is the left inverse of $A$ therefore $LA=I$. Again from $AX=B$, we get $LAX=LB$ ...
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1answer
15 views

show multivarable functions are one-to-one, onto.

$F:\mathbb{R}^3 \rightarrow \mathbb{R}^3, F(x,y,z)=(2x,y,3z+y)$ My current method for these sort of questions is to try to find the matrix that represents this transformation and then see if i can ...
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2answers
47 views

Inverse functions: what is the difference between $\tan^{-1}(x)$ and $\tan(x)^{-1}$?

I’ve never really been taught about inverse functions, and I figured this is a pretty simple question, but I couldn’t find any explanation in my math textbook about this. What is the difference ...
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2answers
44 views

Composition and inverse mappings

Let $A\stackrel{\alpha} \rightarrow B \stackrel{\beta}\rightarrow A$ satisfy $\beta \alpha = 1_A$. If either $\alpha$ is onto or $\beta$ is one to one, show that each of them is invertible and that ...
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2answers
73 views

Derivative of trace of inverse of a matrix function

I am trying to derive the derivative of the trace of inverse of a matrix function (of X), i.e. $$f(X)=Tr\left((HXH^{H}+I)^{-1}\right) $$ where $H\in R^{n\times m}, X\in R^{m\times m}$. So ...
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1answer
29 views

What is the inverse of the function $f: x \mapsto (x,x^{2}) : \mathbb{R} \to \mathbb{R}^{2}$?

Let $f: x \mapsto (x,x^{2}) : \mathbb{R} \to \mathbb{R}^{2}$ and let $Y := f(\mathbb{R})$. Then $\mathbb{R}$ and $Y$ are in injection via $f$. Moreover, since $Y$ is the range of $f$, certainly ...
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2answers
55 views

$(a,b) \mathbin\# (c,d)=(a+c,b+d)$ and $(a,b) \mathbin\&(c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$. Multiplicative inverse?

Let $r\in \mathbb{R}$ and let $0\neq s \in \mathbb{R}$. Define operations $\#$ and $\&$ on $\mathbb{R}$ x $\mathbb{R}$ by $(a,b) \mathbin\#(c,d)=(a+c,b+d)$ and $(a,b) ...
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1answer
22 views

Matrix and eigenvalues question hints?

This is the homework I have done part a, b, but I don t have any idea how to do the rest $y = 5$ and $z = 12 $ Those are the eigenvalues of matrix $A$ For part c, and d, I've tried to put some ...
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23 views

Inverse of the sum of a symmetric positive definite matrix and a diagonal (but with different entries) matrix

Suppose we have symmetric positive definite $A$ with the size of $d \times d$, giving the SVD $A=V\Sigma U^T$ , if $D$ is an identity matrix, ie $D=I$, then $(A^T A + \gamma I)^{-1}=U (\Sigma^2 + ...
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1answer
33 views

Solutions for the dependency problem

Currently I read about the dependency problem of interval arithmetic. Mainly it's the problem that in the equation $X-X$ for $X$ being an interval the following is calculated: $$X-X=\{x-y:x\in X, y\in ...
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3answers
141 views

Geometric interpretation of inverse complex function?

Function $f\colon\mathbb{R}\to\mathbb{R}$ and its inverse $f^{-1}$ are symmetric over line $y=x$. It's easy to imagine inverse of real function, we just have to "flip" the plot over $y=x$. But what ...
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0answers
104 views

If $f^{-1}(x)$ is continuous, is $f(x)$ also continuous?

Let $f:\mathbb{R}\mapsto\mathbb{R}$ be a one-to-one function with $f(\mathbb{R})=\mathbb{R}$. If $f^{-1}(x)$ is continuous $\forall x\in\mathbb{R}$, prove or disprove that $f(x)$ is continuous ...
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1answer
46 views

Inverse sum representation of sine

The other day I was playing with functions of the form $$ f(x) = \frac{1}{\frac{1}{a_0(x-b_0)} + \frac{1}{a_1(x-b_1)} + \cdots + \frac{1}{a_n(x-b_n)}} $$ and I found particularly that $$ ...
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1answer
34 views

Cayley transform a matrix that is invertible when added to the identity

Let A be an nxn matrix such that (I+A) is invertible. I need to prove that the Cayley Transform of A, denoted by $A^c$, is such that $(I+A^c)$ is invertible. The Cayley Transform is defined as ...
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78 views

Geometric Interpretation of Determinant of an Inverse Matrix

The $\mathbf{A}$ be an $n\times n$ full rank matrix. Then, the (signed) volume enclosed by the rows (or columns) of $\mathbf{A}$ is equal to $\det(\mathbf{A})$. My question is, what is a geometric ...
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236 views

Proof of the Inverse Function Theorem using the Contraction Mapping Principle.

I've been set this problem recently and I'm having a lot of trouble with it. Any help would be much appreciated! Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function with continuous derivatives ...
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1answer
54 views

Reciprocal of the reciprocal of zero

By straightforward evaluation, $$(0^{-1})^{-1}=(NaN)^{-1}=NaN$$ where $\frac{1}{0}$ is taken to equal $NaN$ (not a number), or undefined or indeterminate. However, the laws of exponents state that ...
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Is the condition “the inverse image of a closed base set is closed?” sufficient for continuity?

Let's say you have a function $f:X \to Y$, where $X$ and $Y$ have topologies. The set $C$ forms a closed base for $Y$. If for every $c \in C$, $f^{-1}(c)$ is closed in $X$, is $f$ continuous? If the ...
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A class of sparse matrices whose inverse is also sparse?

In general the inverse of a sparse matrix is dense. A notable (but trivial) exception from that rule are diagonal matrices. Is there any other (broad) class of sparse matrices whose inverse is also ...
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1answer
38 views

Finding multiplicative inverse Euler's theroem

been struggling this whole day with trying to figure out the multiplicative inverse of 17 modulo 31 using Eulers theorem. We know that 31 is a prime, φ(n)=30, so i end up with 17^30=(cong)1 (mod 31). ...
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49 views

Is the linear operator $T(f)(x) = f(-x) + f(x)$ invertible?

My understanding is that the inverse of a linear operator will effectively "undo" the operation. Therefore to get the inverse of this operator we need to somehow subtract the f(-x). But I'm not ...
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1answer
36 views

Is the linear operator $T_f(x) = f'(x)$ invertible?

I think that $T_f(x)=f'(x)$ is invertible. This seems likely because it is a differential operator, and the inverse of a differential operator is the integral operator (though I'd like more ...
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1answer
27 views

Condition inverse $p$-adic number

Take $p$ prime, $n \in \mathbb{Z}_{>0}$ and $x \in \mathbb{Z}_p$. Suppose that $p$ isn't a divisor of $$x = (x_j + p^j \mathbb{Z})_{j \in \mathbb{Z}_{>0}},$$ then one can prove that the first ...
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Why does this “miracle method” for matrix inversion work?

Recently, I answered this question about matrix invertibility using a solution technique I called a "miracle method." The question and answer are reproduced below: Problem: Let $A$ be a matrix ...
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Find inverse of 15 modulo 88.

Here the question: Find an inverse $a$ for $15$ modulo $88$ so that $0 \le a \le 87$; that is, find an integer $a \in \{0, 1, ..., 87\}$ so that $15a \equiv1$ (mod 88). Here is my attempt to answer: ...
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3answers
31 views

$A,B$ are matrices $3x3$ so that $B^2A=-2B^3$ and $B^3+AB^2=3I$ express $A^{-1}$ and $B^{-1}$ using $B$

I have the follow question : $A,B$ are matrices $3x3$ so that $B^2A=-2B^3$ and $B^3+AB^2=3I$ express $A^{-1}$ and $B^{-1}$ using $B$ I tried to "play" with the equations but I always get stuck with ...
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1answer
36 views

Understanding Multiplicative Inverse in RSA

Okay so I am reading up on RSA, trying to understand how it works, and I come across this $ x∈ℤp, x−1 ∈ℤp ⟺ \gcd(x,p) = 1$ Now it then gives an example, as follows: Lets work in the set $ℤ9$, ...
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Finding the $h'(x,y,z)$ if $h= p \circ q $ $p(x,y,z)=(x \sin y, x \cos y, z+y ), q(x,y,z)=(x^2,x+y,2e^z)$

I just want someone to check my work basically. Providing thoughts and insight, into possible mistakes: Finding the $$h'(x,y,z)$$ if $$h= p \circ q ,\ \ p(x,y,z)=(x \sin y, x \cos y, z+y ), \ \ ...
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102 views

Inverse of a product of real functions

Given $F(x) = L(x)G(x)$, with $L$ and $G$ real function strictly greater than zero. Suppose that F and G are decreasing functions (so that $F^{-1}$ and $G^{-1}$ exists). What can we say about the ...
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3answers
117 views

Why can't this mixed function be inverted?

Given the function $$y=Ax + B\sqrt x$$ where $A$ and $B$ are real constants, $x$ is real and $x > 0$ I want to find the inverse where $x$ is a function of $y$. ButI don't believe that's possible ...
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0answers
55 views

What is Lebesgue measure of sets of inverse prime numbers in $[0,1] $?

I would like to know if it is possible to know the lebesgue mesure of sets of inverse prime numbers in $[0,1]$ Note : I think should to know in the first if the sets of primes are infinit countable ...
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1answer
58 views

Block matrix pseudoinverse: symmetry of the inverse of a symmetric matrix

In the wiki page for block matrix pseudoinverses, there is a formula $$ \begin{pmatrix}A & B \\ C & D\end{pmatrix}^{-1}=\begin{pmatrix} (A-BD^{-1}C)^{-1} & ...
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2answers
63 views

Inverting a $3\times 3$ block matrix

Suppose that $a$ and $b$ below are scalars, $F$ a square matrix, $v$ a column vector. I'm trying to invert the matrix $M$ of the form $$ M=\begin{pmatrix} a & v' & 0\\ v & F & 0\\ 0 ...
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0answers
22 views

Finding the inverse of a square circulant matrix

I'm having difficulties of finding the inverse of the following square matrix, which consists of $2\times 2$ circulant matrices: $A = \left[\begin{matrix}x^{383} & x^{102} + x^{253} \\ ...
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31 views

Flip Values to get the opposite

Not sure of the name of what I need to do, but I used to do it all the time, and now i forget. I have values 1 - 10. I want 10 to become 1 and 1 to become 10. What is the formula to do this again? It ...
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1answer
23 views

Calculate the modular inverse of $2a$ given that of $a$

My problem is that I have to calculate some modular inverses of numbers that are related by multiplying by $2$, that is: Given $a$ and $x$ so that $ax\equiv1\mod n$ ($n$ being an odd number) I need ...
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3answers
131 views

Can we prove $BA=E$ from $AB=E$? [duplicate]

I was wondering if $AB=E$ ($E$ is identity) is enough to claim $A^{-1} = B$ or if we also need $BA=E$. All my textbooks define the inverse $B$ of $A$ such that $AB=BA=E$. But I can't see why $AB=E$ ...
0
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1answer
128 views

L2 norm of an inverse of a sum of matrices

I am trying to take the L2 norm of the following expression: $-(H^{-1} + bI)^{-1}v$, where $H$ is a psd matrix, b is a scalar, and $v$ is a vector. In particular I am having trouble with the first ...
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3answers
110 views

If a function has an inverse then it is bijective?

I have some trouble finding the answer to this, can someone help me out: If I have a general function $f$ with domain $X$ and codomain $Y$, I know nothing about the function (injective, surjective). ...