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Questions tagged [inverse]

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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34 views

Is $[\sin(1/2)]^{-1}$ identical to $\frac{1}{\sin(1/2)}$

I'm in Grade 12 Advanced Functions and having some trouble with understanding the difference between $\sin^{-1}(1/2)$ and $(\sin(1/2))^{-1}$. I recognize that the former asks to find an angle whose ...
1
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1answer
22 views

Properties of matrix inverse

Suppose $A$ is a $l \times l$ matrix and $\Gamma$ is a $l \times k$ matrix. Is it true that $\Gamma[\Gamma'A\Gamma]^{-1}\Gamma' = A^{-1}$, assuming $A^{-1}$ exists? If yes, how to show it?
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2answers
32 views

How can I find the inverse of a permutation?

My question is, how can the inverse of $5 9 1 8 2 6 4 7 3$ be $3 5 9 7 1 6 8 4 2$? At first glance, 1 and 2 are both less than 3, for example, which seems to conflict with the instruction "then ...
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0answers
37 views

Mathematical topic of functions derivative with respect to own derivative for finding inverse

Is there any topic or branch within mathematics that uses a functions derivative with respect to a different derivative of the same function to solve inverses of functions? E.g. with some function f(...
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2answers
36 views

Inverse of ratio function [on hold]

$$ f(x) =\frac{4x^3}{x^2+1}$$ Question is $$ \frac d{dx} (f^{-1}(2))=? $$ Now i know how to invert exponential function, rational function and etc but i don't understand how can i invert this ...
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2answers
53 views

Inverse of $\ln(e^x-3)$

so the whole concept about inverses is a little foggy. Say you have function $f(x)=\ln(e^x-3)$ and you want to know the inverse function, then: $$\ln(e^x-3) = y$$ $$e^x-3 = e^y$$ $$e^x = e^...
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4answers
56 views

Prove $B^{-1}=I+B-B^2$ [closed]

I've been trying to prove this for so many hours but nothing seems to work. Probably I am just missing something. Anyone has any idea about it? The question is to prove: $B^{-1}=I+B-B^2$ Any ...
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1answer
24 views

Trying to find the inverse of $B$ knowing the optimal solution

Can one know how is $B^{-1}$ and $\left(\matrix{ b_1 \\b_2}\right)$ defined knowing that $c_BB^{-1}b=150$ and $B^{-1}b=B^{-1}$ $\left(\matrix{b_1\\ b_2 }\right)=\left(\matrix{30 \\ 10}\right)$ ? We ...
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1answer
22 views

What's the inverse of the gradient of a gradient?

I am wondering if, for a given functio $f(\mathbf{x})$, there exists a tensor $\mathbf{A}$ such that $$\nabla \nabla f \cdot \mathbf{A} = \mathbf{I}$$ To make it clearer, in index notation (with ...
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2answers
44 views

Inverse of tridiagonal Toeplitz matrix

Consider the following tridiagonal Toeplitz matrix. Let $n$ be even. $${A_{n \times n}} = \left[ {\begin{array}{*{20}{c}} {0}&{1}&{}&{}&{}\\ {1}&{0}&{1}&{}&{}\\ {}&...
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0answers
28 views

inverse function / univariate transformation

I want to find the probability density function (PDF) of $$ y = f(x) = b ~x^2 - c + d \cos(\alpha x) + k~ {\rm sinc}(\alpha x),~~~~\alpha, b, c, d, k > 0 $$ where ${\rm sinc}(\alpha x) = {\sin}(\...
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0answers
18 views

Cheap Implementation for the Moore Penrose Pseudoinverse

Can some one explain Adam W's implementation for finding the Moore Penrose Pseudoinverse? http://math.stackexchange.com/questions/75789/what-is-step-by-step-logic-of-pinv-pseudoinverse/317053#317053 ...
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2answers
446 views

Artin's Algebra Exercise 1.1.16

The question goes $A^k = 0$, for some $k>0$ ($A$ is square). Prove that $A+I$ is invertible. I did $A^k = 0 \implies A^k+I=I$. So, $(A+I)(A^{k-1} +.... + I) = I$. So it's invertible with the ...
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2answers
32 views

Linear Algebra Question. Invertibility Problem.

$A$ is a $d*k$ full rank matrix, given d > k. Is the matrix $I-AA^T$ invertible?
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3answers
67 views

Proving that the inverse of 2 is not an integer

I have been trying to prove that the inverse of $2$ does not exist in integers, but I did not succeed. I tried to assume negatively that is an integer, but I have refrained from moving on... I am not ...
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2answers
48 views

Inverse $4\times 4$ matrix contains left upper $3\times3$ transposed matrix

I am not a matrix geek or something. I just remeber a couple of things from the university math classes. Maybe the explanation is simple. What's going on: I go through a code of a certain game and an ...
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0answers
34 views

Getting the inverse of a matrix with matrix elements

I am solving a problem regarding newton's method. We are using this as the function: click to see function Thus these are the Jacobian Matrix and the set up for Newton's Method: click here to see ...
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6answers
35 views

Inverse modular multiplicative

So this seems really confusing for me. Supposedly, 4-1 mod 5 = 4. Isn't the inverse multiplicative of 4 equal to $\frac{1}{4}$? If so shouldn't $\frac{1}{4}$ mod 5 be equal to $\frac{1}{4}$ ?
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1answer
18 views

Woodbury's formula with block-diagonal matrix

From Woodbury's formula, it is easy to show that \begin{align*} (a \mathbf{I}_n + b J_n)^{-1} = \frac{1}{a}\mathbf{I}_n - \frac{b}{a(a+bn)}J_n \end{align*} where $\mathbf{I}_n$ is the $n\times n$ ...
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0answers
34 views

Question about the inverse of a positive definite matrix? [closed]

So I was reading the book on Graphical Models by Joe Whittaker. And, in chapter 5 of the book, there is a question about showing that the diagonal elements of the inverse of a positive definite matrix ...
0
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2answers
18 views

Finding the reciprocals using trigonometric functions and their inverses

If $x\neq0$, how to find the reciprocal of $x$ only by using trigonometric functions and inverse trigonometric functions? I have found only one answer, which is; $\tan (\cos ^{-1}(\sin(\tan^{-1}(x))))...
2
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1answer
29 views

Prove domain of $f^{-1}$ is $(c,d)$ and the limits of the endpoints of $f^{-1}$ are equal to the enpoints of the domain.

Suppose $f$ is defined, continuous, and strictly monotone increasing on $(a,b)$ such that $$ c = \lim\limits_{x\to a^+} f(x) \text{ and } d = \lim\limits_{x\to b^-} f(x) $$ exist in the extended ...
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3answers
61 views

Matrix inversion to solve least squares problem

I want to solve a least squares problem in the form of $\mathbf{A}\vec{x}=\vec{b}$, where $\mathbf{A}$ is a $m\times n$ matrix asociated to the transformation $T:\mathbb{R}^n \to \mathbb{R}^m$; and ...
2
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2answers
28 views

Mapping values in the range [-1, 1] to [0, 1] in an invertible fashion

I have a continuous variable whose range is within $[-1, 1]$. I want to map the values of this variable to the range $[0, 1]$ instead. What I do is I add the value of $1$ to the the variable and ...
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0answers
16 views

Linear Algebra Inverse Problem [duplicate]

I am studying Linear Algebra by myself and I have nowhere to ask.. Please help me T_T Given that AX=I for 3x3 matrix A and B, can I conclude that A is invertible and the inverse of A is X? I'm not ...
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5answers
97 views

Let $A_{n\times n}$ be a real matrix. Is it true that $I+A^TA$ is invertible?

Let $A_{n\times n}$ be a real matrix. Is it true that $I+A^TA$ is always invertible?
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0answers
18 views

Inverse function of 2D-circle: f(u,v,a) = (x,y,r)

Can someone explain to me, how I can get $f^{-1}(x,y,r)=\begin{bmatrix}u\\v\\a\end{bmatrix}$ Someone told me to look into the inverse function theorem, but I only have high-school education, I know ...
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1answer
21 views

How to find the inverse/unit for mod?

So for this problem, its asking to find the inverses of [4], [5], and [7]. I'm trying to skim through my textbook but I can't find an example. Like for this problem, I don't understand why the Units ...
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3answers
29 views

Inverse by left multiplication but not right?

Suppose $T: V \rightarrow W$ and $U: W \rightarrow V$ are linear transformations. It is known that $ U = T^{-1}$ if $UT = I_V$ and $TU = I_W$. Is it possible to then also have a transform $Z: W \...
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0answers
31 views

How to calculate the inverse of a Hessian matrix?

Hello does anybody know how to calculate the inverse of a Hessian matrix ? A link to a solved example would be awesome. Thanks for any advice.
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0answers
20 views

Vandermonde infinite matrix inverse

I am searching for an inverse of a certain infinite matrix, Vandermonde one. I have been searching in bibliography and some well known examples exist in literature: Pascal Matrix Inverse -> ...
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1answer
19 views

Matrix inversion properties

I would like to know if there exist any identity regarding to the inverse of a matrix in which a row or (or a column) has been multiplied by a constant. I know that if whole matrix is multiplied, ...
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1answer
27 views

Prove the derivative of $\arctan(x)$ using derivative definition

Prove that the derivative of $\arctan(x)$ is $\frac1{1+x^2}$ using definition of derivative I'm not allowed to use derivative of inverse function, infinite series or l'Hopital.
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1answer
40 views

Is there a name for this particular type of matrix?

Consider the following matrix structure: $$ M = \begin{pmatrix} a & d & c & c \\ d & a & b & b \\ c & b & a & e \\ c & b & e & a \end{pmatrix} $$ It ...
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2answers
51 views

Proving the inverse of $1/(1 + 2x)$ is $(1 - y)/2y$

Let $f(x) = 1/(1 + 2x)$. I want to show that $f^{-1}(y) = (1 - y)/2y$. Is it sufficient for me to show $$f\left(\frac{1-y}{2y}\right) = \frac{1}{1 + 2(1 - y)/2y} = \frac{1}{1 + (1 - y)/y} = \frac{1}{...
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2answers
31 views

Computing the image of $(0, 1)$ under $f(x) = 1/(1 + x)$

I would like to compute the image of $(0, 1)$ under $1/(1 + x)$. I know that the answer is $(1/2, 1)$, but I would like to prove it. I am given that the function is strictly decreasing on the ...
0
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1answer
42 views

How to make squaring method of finding an inverse function to be invertible?

I've been trying to find an inverse of this function $$f(x) = e^{-\left(\displaystyle \frac{x}{\sqrt{1-x^2}}\right) \displaystyle \pi }$$ These are the approaches First approach using squaring ...
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2answers
40 views

Show that $\forall x\in \Bbb{R}$ and for a fixed $a\in \Bbb{R},\;t_a= a+x$ is invertible

Let $a\in\Bbb{R}$ be fixed. Define \begin{align} t_a:\,&\Bbb{R}\to \Bbb{R}\\ &x\mapsto a+x\end{align} I want to show that $t_a$ is invertible. Below is my trial MY WORK Suppose $t_a v=0$, ...
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0answers
13 views

How to combine $V$($3\times 3$ matrix) from $V^{-1}XV$?

$V_i$ and $X$ are $3\times 3$ matrices. $$V_k-V_{k-1}=V_k^{-1}XV_k$$ How to get $V_k$ if $V_{k-1},X$ are known? The Problem it is continous system, i discrited that but how to get the $$R_b^l$$ only?...
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1answer
65 views

What is the mathematical approach of inversing a function resulting in a piece-wise solution?

I've been trying to find the inverse of $$f(x) = e^{-\left(\displaystyle \frac{x}{\sqrt{1-x^2}}\right) \displaystyle \pi }$$ Here are my steps $$ \begin{align} x & = e^{-\left(\displaystyle \frac{...
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1answer
20 views

A guess about invertible sign matrix

For any $n\times n$ invertible matrix $A=(a_{ij})$, which satisfies $a_{ij}\in\{0,1,-1\}$. Assume $A^{-1}=(b_{ij})$. I guess $\vert b_{ij}\vert \le 1$, because I have found it is right when $n=1,2$. ...
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1answer
17 views

Indicate functions that have an inverse, that is defined for all reals

My brain is refusing to think about these any more. Apparently my thinking is super flawed here. Functions are: 1) $x^3+x$, 2)$x^3-x$, 3)$\sin x + \sin 2x$, 4) $x + \text{arctg}x$, 5)$e^{\ln{(x^2+1)}}$...
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0answers
47 views

Inverse of a 3x3 block matrix

I would like to get the inverse of a 3x3 (covariance) block matrix \begin{bmatrix}A&B&C\\B'&D&E\\C'&E'&F\end{bmatrix} where the prime ' indicates the transposition operator. ...
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2answers
72 views

Why is $\arcsin(x) = \pi$ an impossible equation?

Why is $\arcsin(x) = \pi$ an impossible equation, if $\sin(\arcsin(x)) = \sin(\pi)$ for $x = \sin(\pi)$?
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1answer
31 views

Problems with finding inverse by using gcd

I´ve tried to solve this problem, but i think i need some help from you guys. In my textbook it is written that the inverse of 35 mod 3 is 2, i get that the inverse is 2, but how do you find it by ...
2
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1answer
86 views

product of two Jacobi matrices

Let $$ A = \begin{bmatrix} a_1 & b_1 \\ b_1 & a_2 & b_2 \\ & b_2 & \ddots & \ddots \\ & & \ddots & \ddots & b_{n-1} \\ & & & b_{n-1} & a_n \end{...
0
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3answers
72 views

Inverse function to f

I have function f : $\mathbb{R}$ → $\mathbb{R}$ defined by f(x) = $e^{-3x}-3e^{-2x}$ and have found that f'(x)=$-3e^{-3x}+6e^{-2x}$. Can someone explain why f does not have an inverse function. And ...
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0answers
30 views

Probability that the sum of $k$ matrices is invertible

Suppose we have $k$ matrices over $\mathbb{Z}_q$ of size $n \times n$, where $q \gg k$ and the entries of the matrices are chosen uniformly at random. Assume $q$ to be prime and we do not pick any ...
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1answer
33 views

Inverse of a certain unit upper triangular matrix

I don't know if there is a certain name for this matrix. but I want to show $\begin{pmatrix}1&\gamma&\gamma^2& \ldots & \gamma^n\\ &1&\gamma&\ddots&\vdots\\ &&\...
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1answer
27 views

Showing that the generalized inverse for a square invertible matrix is unique

Let $A\in\mathbb{C}^{m\times n}$, then a generalized inverse matrix $B$ of $A$ satisfies the following $$ABA = A \ \text{and} \ BAB = B.$$ I am to show that $B$ is unique if $A$ is square and ...