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.

learn more… | top users | synonyms

0
votes
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 $f^{-1}(A)\cap f^{-1}(B)$?
2
votes
0answers
25 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 ...
0
votes
2answers
44 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?
1
vote
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 \mathbb{R}...
0
votes
3answers
21 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 ...
0
votes
3answers
22 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 ...
1
vote
0answers
15 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 &...
2
votes
1answer
77 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 \\...
1
vote
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 ...
3
votes
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 $a^p \equiv a^0 \equiv 1 \pmod p$ ?
0
votes
0answers
17 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.
5
votes
2answers
45 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 v)=\...
0
votes
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 ...
0
votes
1answer
48 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 \mathscr{...
4
votes
3answers
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?
0
votes
2answers
44 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) ...
0
votes
0answers
25 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 [0,\...
0
votes
1answer
23 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 ...
4
votes
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 $b = x^{-1}$, ...
0
votes
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
1
vote
2answers
81 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 ...
0
votes
2answers
37 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 $T\...
1
vote
2answers
72 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.
2
votes
2answers
22 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 ...
2
votes
5answers
81 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 ...
3
votes
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 (...
1
vote
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 ...
0
votes
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 =...
0
votes
1answer
26 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 ...
0
votes
1answer
23 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& ...
0
votes
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 = p_1^{\...
0
votes
1answer
14 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?
1
vote
2answers
25 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
vote
1answer
46 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 ...
1
vote
1answer
50 views

Inverse function to $f(t)=3t+4ln(t+1)=y$

I have to invert the function $f(t)=3t+4\ln(t+1)=y$, so $f^{-1}(y)=t$. But I am struggling to invert this. Is there a solution?
0
votes
1answer
72 views

Prove $sgn(π) = sgn(π^{-1})$?

I'm pretty sure the inversion count of $π$ should be the opposite of the inversion count of $π^{-1}$. By this I mean if $π$ looks like this: $1 \to 1$, $2\to 2, \ldots, 10 \to 10$ and therefore the ...
0
votes
2answers
21 views

Simplifying Inverse Trig Function

I'm trying to figure out how to simplify this expression but I'm not quite sure on how to approach this question. How should I approach this question? Any help is greatly appreciated! $\tan(\sin^{-1}(...
3
votes
1answer
26 views

Show that a matrix satisfying certain conditions is non-singular

I have a square matrix $A$ satisfying the following conditions: The elements on the diagonal are negative; All other elements are non-negative; All row sums are less than or equal to $0$; There is ...
0
votes
1answer
26 views

I need someone to show me how to solve this input/output problem

Alright, so I have: $4y^3 = x$ And now I have to solve for $y$, where I can later use that equation to answer other questions I have. Can someone hint me out on how to solve for $y$ given the above ...
0
votes
0answers
18 views

The existance of Schur Complement Inverse

A block matrix $\mathbf{M}=\left[ \begin{array}{ccc} \mathbf{A} & \mathbf{B} \\ \mathbf{B}^T & \mathbf{C} \end{array} \right]$ is invertible if $\mathbf{A}$ and $(\mathbf{C}-\mathbf{B}^T\...
0
votes
3answers
39 views

Is the inverse of a real, continuous “1-1” function necessarily continuous itself? [closed]

If so, please do provide me with an epsilon-delta proof, if possible. Thanks in advance.
0
votes
1answer
21 views

Inverse of the sum of a invertible matrix with known Cholesky-decomposion and diagonal matrix

I want to ask a question about invertible matrix. Suppose there is a $n\times n$ symmetric and invertible matrix $M$, and we know its Cholesky decomposion as $M=LL'$. Then do we have an efficient way ...
0
votes
0answers
14 views

Harmonic inversion of an eccentric circle.

Inverted here is a circle with respect to another circle not as the conventional reciprocal inversion $ r_1 = \dfrac{a^2}{r_2}, $ but by means of a Lens formula known from time of Gauss: $$ 1/r_1 + 1/...
0
votes
1answer
20 views

Related to symmetric, diagonal and invertible matrices

While solving a problem I came across a specific question: Given $A,B$ as $2$ real, symmetric, matrices with $B$ positive definite, does there exist a matrix (invertible) $P$ such that both $P^TAP$ ...
0
votes
0answers
35 views

Finding the inverse of linear transformation using matrix

Assuming I have a linear transformation represented by a matrix with respect to some random bases, how could I find the inverse of the transformation using the matrix representation? I know I should ...
0
votes
2answers
24 views

Congruent matrices - why do we require invertiblility?

If $A$, $B$ $\in K^{n \times n}$ are $n \times n$ matrices over a field $K$, then we say that $A$ and $B$ are congruent if there exists an invertible $P \in GL(n, K)$ such that $B = P^TAP$, where $P^T$...
2
votes
0answers
55 views

Derivative of $(\lambda I - A)^{-1}$ with respect to $\lambda$

Is need to work with $\frac{d}{d\lambda} (1 - v^{T}(\lambda I - A)^{-1}u)$. Is it true that: $$\frac{d}{d\lambda} (1 - v^{T}(\lambda I - A)^{-1}u) = -v^{T}\frac{d}{d\lambda}(\lambda I - A)^{-1}u$$ ...
1
vote
0answers
15 views

How can I convince myself of the Fourier scaling property via inverse FT?

I have this function $f(at)$, and I want to Fourier-tranform it. I proceed in the following way, for $\quad\alpha<0 \Longrightarrow a=-|a|$: \begin{align} \ \mathcal{F}_{t \rightarrow \xi}[f(at)]= ...
0
votes
0answers
19 views

Find $v$ that maximises $v^TA\left(I_m-\sum_{i=1}^{p}B_i^Tvv^TB_i\right)^{-1}A^Tv$

I am trying to generate a rank 1 update rule for an optimization problem and I reach a point where I don't know how to continue. Let $v\in \mathbb{R}^n$ such that $\|v\|_2=1$ and also consider that $...
2
votes
1answer
41 views

Is there a way to update the inverse of a sum of two matrices following a rescaling of one of them?

Suppose I have two matrices $A$ and $B$ (let's assume that both $A$ and $B$ are invertible, as is their sum), and a scalar $g$. I am interested in the matrix $$M^{-1} = (A + gB)^{-1}$$ I am aware ...