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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19 views

Relation between powers of inverse modulo n.

Recently, I was studying enchanced euclidean algorithm. I am wondering if there is some way to calculate inverse of $a^2$ (and higher powers) modulo $n$, knowing inverse of $a$ modulo $n$. For ...
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3answers
40 views

Value of the given expression …

If $$y=\tan^{-1}\left(\sqrt{\dfrac{1+\cos x}{1-\cos x}}\right)$$ then value of $(2x+14y)^3-343$ is ? I reduced the equation as $y=\tan^{-1}\left(\dfrac{1+\cos x}{\sin x}\right)$ but I couldn't ...
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1answer
21 views

Using inverse of transpose matrix to cancel out terms?

I am trying to solve the matrix equation $A = B^TC$ for $C$, where $A$, $B$, and $C$ are all non-square matrices. I know that I need to utilize $M^TM$ in order to take the inverse. I'm just not sure ...
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1answer
48 views

Differentiation Involving Determinant.

I have to compute the following differentiation : $$\frac{\partial}{\partial\sigma^2}\det[\mathbf X_{p\times n}'(\sigma^2 \mathbf I_{n}+\mathbf Z_{n\times q}\mathbf G_{q\times q}\mathbf Z_{q\times ...
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1answer
29 views

If a linear eq. System has only a single solution, the matrix has always an inverse? [on hold]

If i have a 3x3 matrix of a linear eq. System and i archieve an upper triangular matrix (means unique solution) does this form always have inverse matrix? Thank you very much!
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2answers
116 views

If $f^{-1}(x)=\frac{1}{f(x)}$ then find $f(1)$

For $a>1$ we have: $f:[\frac{1}{a},a]\to [\frac{1}{a},a]$ be a bijective function. Suppose $f^{-1}(x)=\frac{1}{f(x)}$ for all $x \in [\frac{1}{a},a]$ then find $f(1)$. Could someone give me ...
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0answers
36 views

discrete random variable with uniformely distributed random variable

I hope you can help me because I have no clue where to start: Let X be a discrete random variable with $ p_k=P_X[X=x_k]=p(x_k) $for all $1\le k\le N$ for $N\in \Bbb N$ and distribuition function: ...
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1answer
61 views

Function inversion (analytical)

Can $t(x)$ be found from: $$A \, t + B\ln\frac{1-t}{t}=x \; ?$$ Here, $A>0, \; B < 0$ and $0 \lt t \lt 1$. The $t(x)$ should be given in analytical form (even if you use, say, Lambert's W - ...
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1answer
22 views

Using the Affine cipher, do we need $a^{-1}$ if we know gcd(a,26)=1?

I have just attempted the affine cipher with the word "code" $CODE = 02140304$ Lets choose our key as $(5,3)$, so our encryption is $y=5x+3$ $13211823=NVSX$ Now, to undo the code, I would have to ...
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1answer
80 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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1answer
59 views

Inverting an $n \times n$ matrix using determinant

We're asked to invert the following matrix with the help of guided questions. $$\begin{pmatrix} 1 + a_1 & 1 & \cdots & 1 \\ 1 & 1+a_2 & \ddots & \vdots \\ \vdots & \ddots ...
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2answers
94 views

Why can't the inverse of $F(x)= x+\sin(x)$ have a formula algebraically?

I'm only curious why the inverse of $f(x)$ can not be determined algebraically. Is it because the inverse of $\sin(x)$ cannot be converted into a formula?
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1answer
41 views

Finding the inverse Laplace transform of this function

Find the inverse Laplace transform of this function (related to my question earlier): $$f(t)=\mathcal{L}_s^{-1}\left[\frac{s}{s+\frac{1}{\tau}}\cdot\frac{A}{s}\left(1-\mathrm ...
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0answers
21 views

Inverse trig Function

Hi mathematicians some few questions regarding inverse trig functions I just want to know whether I'm on the right track since this topic is basically new to me 1.) $\arccos (\sin 12 \pi /11)$ ...
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2answers
49 views

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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3answers
20 views

Inverse of a function on two sets.

I understand that $f^{-1}(A\cup B) = f^{-1}(A)\cup f^{-1}(B)$, but what is $f^{-1}(A\cap B)$? Is it necessarily $f^{-1}(A)\cap f^{-1}(B)$?
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0answers
23 views

Perturbations to a matrix causing drastic changes to matrix inverse.

I'm reading this article about matrix norms because I want to understanding the math behind SVD. One of the interesting issues it brings up quite soon is the effect of perturbations to a matrix on ...
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2answers
42 views

Finding modular inverse of every number mod 26?

I am looking at cryptography, and need to find the inverse of every possible number mod 26. Is there a fast way of this, or am i headed to the algorithm every time?
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1answer
41 views

Formal Notation for Finding Inverses of Functions

Generally in most introductory university courses, finding the inverses of functions, is done in what seems to be to be a very haphazard way. Given any scalar function $f : \mathbb{R^n} \to ...
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3answers
20 views

What is the Order of operations for finding the inverse of a function AND solving.

I have $y=4(x+2)^3$. So first part of taking the inverse is switching the variables $x$ and $y$ so you'd have $x=4(y+2)^3$. Why does the exponent $3$ get put in front of the square root symbol? The ...
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3answers
21 views

Proving facts about the inverse of a matrix

Let A and B be matrices. Show that: $(A^{-1})^{-1} = A$ $(A^{T})^{-1} = (A^{-1})^{T}$ $(AB)^{-1} = B^{-1}A^{-1}$ I think I'm supposed to use the inverse property (That $AA^{-1} = I$, where I is ...
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0answers
13 views

Band Matrix w/ real Diagonal and imaginary Off-Diagonal: Structure of Inverse

I am seeking an exact justification for the following property. Consider a symmetric band matrix with real diagonal and imaginary off-diagonal. $$\left(\begin{array}{ccccc} a_1 & ib_1 & 0 ...
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1answer
76 views

Inverse of 2 by 2 matrix verification

I have worked our the solution to a problem, but I want to explain the solution in a mathematical way. I have the following matrix: $$ \begin{pmatrix} 1 & 2 \\ 1 & 1 ...
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0answers
21 views

Will this function be odd?

Question: If $f:R\to R$ is an invertible function such that $f(x)$ and $f^{-1}(x)$ are symmetric about the line $y = -x$, then: A) $f(x)$ is odd B) $f(x)$ and $f^{-1}(x)$ may not be ...
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4answers
65 views

Fermat's little theorem question: why isn't $a^p \equiv 1$?

Fermat's little theorem says that $a^p \equiv a \pmod p$. I have kind of a stupid question. Since $p \equiv 0\pmod p $, why isn't $a^p \equiv a^0 \equiv 1 \pmod p$ ?
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0answers
9 views

Inverse normal CDF formula

Why there is no formula for the inverse normal cumulative function? It has been a while since I studied math so any help would be appreciated.
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2answers
41 views

Invertibility of a linear transformation without knowing its matrix

Let $\mathbb{V}$ be a finite-dimensional inner product space, and let $\mathbb{W} \subset \mathbb{V}$ be a subspace. Define $T:\mathbb{V} \rightarrow \mathbb{V}$ by $$T(\overrightarrow ...
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0answers
22 views

solution of matrix equation

I was trying to solve the problem I have posted previously (here). and stuck up at the point where I need to find a simplified expression for $(\mathbf{I-DW})^{-1}$ Where $\mathbf{W}$ is a doubly ...
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1answer
18 views

Knwing when the inverting operation were wrong with $A^{-1}A$ result

I don't know why but I'm really really weak in inverting matrices since years... I always do some mistakes. I'm asking you how could I cope with that problem and be able to invert matrix easily in the ...
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1answer
43 views

differentiation of a norm of matrix function

I need to differentiate the following function W.r.to $x$ $y=\|x (\mathbf{I-W}-x \mathbf{Diag(v_2)W})^{-1}\mathbf{v_1} - b\|_2$ where $0<x<\frac{2}{max_i{|{v_2}_i|}}$,$\mathbf{v_1}\in ...
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3answers
132 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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2answers
40 views

AP Calculus BC - Derivative of inverse problem

Let $g(x)$ be the inverse of the function $f(x)$. Given the following values on the table below, at which value $x=a$ will $g'(a)=1/6$? (No calculator allowed) ...
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22 views

Inverse of a function and Inverse function theorem.

Apologies if this question is too primitive for professionals here. I understand the inverse of a function, in terms of domain, co-domain and bijections. Let's say $f,g:[0,\infty)\rightarrow ...
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1answer
20 views

When diagonalizing a matrix, in what order should you arrange the the eigenvectors to form the invertible matrix $P$?

I was following this example online to diagonalize a matrix. It lists the eigenvectors as $\lambda =3,2,4$ (note the order). It then arranges each eigenvalue's corresponding eigenvector (3 column ...
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3answers
53 views

If a function maps an input to its inverse, is it bijective?

I read in my textbook that a function is a bijection if and only if it has an inverse. Is it the same thing to say a function $f: X → X$ is a bijection if $f(x) = x^{-1}$? If $a = x$ and $b = x^{-1}$, ...
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0answers
12 views

how to calculate cumulative distribution function inverted exponential in this pic

how to calculate cumulative distribution function inverted exponential in this pic enter image description here
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2answers
80 views

When does $A^TAX = A^TB$?

The original question is a true-or-false question: Assume $A$ is a $m\times n$ matrix and $B$ is a $m\times p$ matrix. If $X$ is an $n\times p$ unkwown matrix, then the system $A^TAX = A^TB$ always ...
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2answers
35 views

Inverse of a square block matrix

I am trying to understand how to compute the inverse of a square block matrix defined as follow: $\begin{bmatrix}2{\bf I}&-{\bf X}\\{\bf X}'&{\bf 0}\end{bmatrix}$, where ${\bf I}$ is a ...
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2answers
70 views

Prove that if $\|A\|<1$, then $(I+A)^{-1}=I-A+A^2-A^3+\cdots$. [closed]

Prove that if $\|A\|<1$, then $(I+A)^{-1}=I-A+A^2-A^3+\cdots$. I'm not sure how to do this. I know the result for $(I-A)^{-1}$, but that won't help me.
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2answers
21 views

Inverse of a matrix with uniform off diagonals

Suppose that we have an all positive matrix where the off diagonal elements are all identical. Can one calculate the inverse of the matrix analytically, or more efficiently than the general case? For ...
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5answers
79 views

Solving $\arcsin\left(2x\sqrt{1-x^2}\right) = 2 \arcsin x$

If we have $$\arcsin\left(2x\sqrt{1-x^2}\right) = 2 \arcsin x$$ we have to find the set of $x$ for which this is true. I tried to solve it by putting $x = \sin a$ or $\cos a, but got no ...
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3answers
70 views

If $f$ function then $f^{-1}$ function iff $f$ function injective (one-to-one).

During the lecture we learned this phrase: "If $f$ is a function then $f^{-1}$ is a function iff $f$ is injective (one-to-one)." But why? What with onto? $f$ doesn't need to be Surjective ...
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2answers
52 views

Solving $\arcsin(\sqrt{1-x^2}) +\arccos(x) = \text{arccot} \left(\frac{\sqrt{1-x^2}}{x}\right) - \arcsin( x)$

If we have to find the solutions of equation $$\arcsin(\sqrt{1-x^2}) +\arccos(x) = \text{arccot} \left(\frac{\sqrt{1-x^2}}{x}\right) - \arcsin( x)$$ Using a triangle I rewrite it as $$2 \arctan ...
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0answers
41 views

Let A=$\tiny\begin{pmatrix}1&1&1\\1&2&2\\ 1 & 2 &3 \end{pmatrix}$ and B=$\tiny\begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 &0 \\ 1 & 1 &1 \end{pmatrix}$

Then (A) there exists a matrix C such that A = BC = CB (B) there is no matrix C such that A = BC (C) there exists a matrix C such that A = BC, but A $\neq$ CB (D) there is no matrix C such that A ...
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1answer
24 views

Laplace transform and inverse laplace transform

1- Find laplace transform for $4e^2t-3\cos^2(2t)+2\cosh(3t)$ My answer $L(4e^2t-3cos^2(2t)+2cosh(3t))=4L(e^2t)-3L(cos^2(2t))+2L(cosh(3t))$ $=\frac4 {s-2}-3L(\cos^2(2t))+\frac{2s}{s^2-9}$ But how ...
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1answer
21 views

symetric matrix inverse

Is there an easy way to invert a 3x3 symmetric matrix? for example A = $\begin{pmatrix} -1& 2& 0\\ 2& -5& 0\\ 0& 0& ...
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1answer
29 views

Verifying multiplicative inverses of modulo n are the elements that are relatively prime to n

A proposition in my book states: $(\mathbb{Z}/n\mathbb{Z})^{\times} = \{a \in \mathbb{Z}/n\mathbb{Z}~|(a,n) = 1\}$ which I want to prove. I start by defining $a$ in terms of prime factors $$a = ...
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1answer
12 views

Find the inverse of $θ:P(\Bbb{Z})→P(\Bbb{Z})$ defined as $θ(X) = \bar X$

Find the inverse of $θ:P(\Bbb{Z})→P(\Bbb{Z})$ defined as $θ(X) = \bar X$ (the complement of $X$)? Would the inverse of the function just be the function itself?
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2answers
23 views

Changed codomain of inverse trigonometric functions

If codomain of $\arcsin(x)$ is $(\pi/2 , 3\pi/2)$ and codomain of $\arccos(x)$ is $(\pi , 2\pi)$ then what should be $\arcsin + \arccos$ equal to ? I thought of putting $x = \sin \theta$ But then ...
1
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1answer
43 views

Inverse of a “Vandermonde-like” matrix composed of power series

Is there an analytical formula for the inverse of a complex matrix whose elements are sets of "power series" except the last term is scaled? Let $0<x_1<x_2<...<x_n$ be monotonically ...