Inversion is the process of creating the opposite. Familiar examples include multiplicative inverse $2 \mapsto 1/2$, inverting functions $f(x) \mapsto f^{-1}(x)$, matrix inverse $M \mapsto M^{-1}$ etc. Please include an additional subject tag such as (linear-algebra) or (arithmetic) to help clarify ...

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5
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129 views

Let A be a square matrix such that $A^3 = 2I$

Let $A$ be a square matrix such that $A^3 = 2I$ i) Prove that $A - I$ is invertible and find its inverse ii) Prove that $A + 2I$ is invertible and find its inverse iii) Using (i) and (ii) or ...
0
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2answers
20 views

Finding a matrix inverse when an equation involving it is a multiple of the identity matrix

Say you had a matrix $A$, and you did an equation like $A^2 - A$, and proved that it was a multiple of $I$. How could you find $A^{-1}$ in the form $rA + sI$ after proving that? I want to do it ...
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0answers
24 views

Gaussian Elimination vs matrix inversion [on hold]

Why Gaussian Elimination is better than matrix inversion in therms of FLOPS? Also how LU decomposition improves the shifted inverse power method?
0
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1answer
32 views

What is meant by In-Place Matrix Inversion?

I come across the term "In Place Matrix Inversion" a lot in numerical libraries like NumPy and ND4J. What does it mean ? How is it different from the normal matrix inversion ? What are the advantages ...
0
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1answer
28 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 ...
1
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0answers
23 views

Matrix Inverse as Series

I am looking for different representations of the inverse of a matrix as a power series. One obvious candidate is the Von Neumann series which is given $$A^{-1} = \sum_{k=0}^{\infty} (I-A)^k$$ ...
1
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2answers
54 views

Solve equation of inverse functions

I have two different functions $y_1=f_1(x)$ and $y_2=f_2(x)$, both invertible but quite complex. I am able to find their inverse functions numerically, i.e. $f^{-1}_1(x)$ and $f^{-1}_2(x)$, by ...
2
votes
1answer
33 views

Inverse of the composition of two functions

If I have a composition of two functions: $$y = f(g(x),h(x))$$ where both $g(x)$ and $h(x)$ are readily invertible, can I find the inverse of the composition? i.e.: Can I find $x = f^{-1}(y)$? I ...
2
votes
1answer
56 views

In which cases are $(f\circ g)(x) = (g\circ f)(x)$?

I have found three cases: 1) If $f$ and $g$ are the same function. 2) If $f$ and $g$ are mutually inverse. 3) If both are polynomials of degree $1$ Maybe there are more.
4
votes
4answers
295 views

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 = ...
1
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1answer
33 views

Comparing matrix norm with the norm of the inverse matrix

I need help understanding and solving this problem. Prove or give a counterexample: If $A$ is a nonsingular matrix, then $\|A^{-1}\| = \|A\|^{-1}$ Is this just asking me to get the magnitude of ...
0
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2answers
50 views

Find the inverse $\dfrac{x}{\|x\|}$ in $\mathbb{R^2}$

I wish to find the inverse of $\dfrac{x}{\|x\|}$, where $x \in \mathbb{R}^2$ Let's do this. Let $$y_1 = \dfrac{x_1}{\sqrt{x_1^2+x_2^2}}$$ $$y_2 = \dfrac{x_2}{\sqrt{x_1^2+x_2^2}}$$ Then $$y_1 = ...
0
votes
2answers
26 views

How to simplify inverse trigonometric function

How to simplify the following equation: $$\sin(2\arccos(x))$$ I am thinking about: $$\arccos(x) = t$$ Then we have: $$\sin(2t) = 2\sin(t)\cos(t)$$ But then how to proceed?
1
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2answers
43 views

Proving matrix properties: [closed]

Prove: (i) $A(I+BA)^{-1}=(I+AB)^{-1}A$ (ii) $(I+AB)^{-1}=I-A(I+BA)^{-1}B$ (i) Consider $A(I+BA)=(A+ABA)=(I+AB)A$ Taking inverse on both sides (invert) ...
1
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1answer
18 views

Generate random variate using inverse transform technique of $ f (x) =a (1+|x-2|)$

I need to generate a random variable with density function: $$ f(x)= \begin{cases} a (1+|x-2|) , & {-1 \le x \le 4} \\ 0, & \text{elsewhere} \end{cases} $$ For that I need to inverse the ...
1
vote
2answers
29 views

Asymptotes of $\arctan (2x)$

My book tells me the horizontal asymptotes of $\arctan2x$ is either at positive or negative $\frac{\pi}{2}$, yet the vertical asymptotes of $\tan2x$ occurs at positive or negative $x=\frac{\pi}{4}$, ...
0
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3answers
26 views

Why does $\frac{1}{6e^{2y}}=\frac{1}{2x-8}$ in this context?

This is the context: I tried substituting $y=3e^{2x}+4$ into $6e^{2y}$but I wasn't able to go any further. Does anyone what exactly is being done in the last step?
0
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1answer
13 views

Geometric progression with reverse order

I have the following problem: Find three positive numbers which have the sum of $70$ and create a Geometric progression ($q>0$, increasing). Their inverse sum equals to $4/70$. Thank you!
2
votes
3answers
50 views

Can you inverse a funcion by rotating it?

In school i sometimes run on some excercises where you need to calculate something that has an inverse function in it but you cannot find the inverse and you need to work your way around it. I know ...
0
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2answers
49 views

Is a factorable polynomial invertible?

The reason there exists no quintic formula that finds the roots of a quintic polynomial is simply because some quintic polynomials are irreducible. But reducible quintic polynomials may be invertible ...
5
votes
1answer
529 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 ...
-4
votes
1answer
61 views

What is the inverse function of $y=x^2 + 3x +2$? [closed]

What is the inverse function of $f(x)=x^2 + 3x +2$? Please show your solution method and demonstrate that $f(f^{-1}(x))=x$
1
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1answer
18 views

Invariant under $x \rightarrow 1/x$?

I started thinking on the following problem. I am interested in finding complex functions of a complex variable such that $\phi(z)=\phi(z^{-1})$ So far, all I could come up with was a family of ...
1
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3answers
190 views

Finding the inverse of a mapping that can be defined as a function on a specific domain

Let $A = \{x \in \mathbb R\mid x\geq2\}$ and $B = \{x \in \mathbb R\mid x\geq1\}$ and the function $f : A\rightarrow B$ is defined by $f(x) = x^2-4x+5$. With this domain and codomain, the function ...
4
votes
1answer
28 views

Does $\sin^{-1}x$ has a vertical tangent

I read that the function $f(x)$ has a vertical tangent at $x=a$ in the domain of $f$ if $$f'(a^-) \to +\infty$$ and $$f'(a^+) \to +\infty$$ Or both approach to $-\infty$. But for $f(x)=\sin^{-1}x$ ...
3
votes
2answers
41 views

For which values of $a,b$ is the matrix invertible?

I am trying to figure out the below question: 15. For which values of the constants $a$ and $b$ is the matrix $$A = \left[\begin{array}{cc} a & -b \\ b & a \end{array}\right]$$ ...
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0answers
15 views

Integral inversion

Say I know this function $$ F(u) = \int _{-\infty}^{\infty}f(x) m\left(\frac{u}{x}\right) \mathrm d x$$ where $m(x)$ is a Fourier transform of an infinitely differentiable real function, whose maximal ...
0
votes
1answer
66 views

Getting $B$ from $A = M^t B M$ without inverting $M$

I have got three matrices: $A$ (dimension $n \times n$), $B$ (dimension $m \times m$) and $M$ (dimension $m \times n$). We have $m > n$. This is the relation between these three matrices: $A = M^t ...
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2answers
48 views

condition number of matrix plus constant times identity

I saw this post on the eigenvalues of a matrix plus a constant times the identity matrix. Say $A$ is an $n\times n$ matrix (real and non-singular) with eigenvalues $\lambda_1,\ldots,\lambda_n$, then ...
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2answers
36 views

Can the cross product of two non-invertible matrices be invertible?

To put it better, if A and B are non-invertible matrices (for whatever reason), can the matrix AB be invertible? Just used to help understand a Linear Transformation assignment question, don't ...
0
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2answers
29 views

Inverse image of the set $[−1, 4)$ under $f : x \mapsto -x^2$

I have an issue with determining the inverse image of a set. I cannot understand anything about it. I've got a simple exercise here, could someone here show me how the inverse image works and more ...
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1answer
36 views

What is $\ln(e^x -4) $, solving for the inverse?

What is $\ln(e^x -4) $, solving for the inverse? I know $\ln(e^x)$ is just $x$, but I don't know what to do with the 4.
0
votes
2answers
685 views

Inverse functions and tangent line

Let $f(x) = \frac14x^3 + 12x + 6$ and let $y = f^{-1}(x)$ be the inverse function of $f$. Determine the $x$-coordinates of the two points on the graph of the inverse function where the tangent line is ...
0
votes
2answers
4k views

Write the following in the form of AX = B

Write the following system of equations in the form $AX = B$, and calculate the solution using the equation $X = A^{-1}B$. $$2x - 3y = - 1$$ $$-5x +5y = 20$$ I'm not the strongest at linear algebra ...
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0answers
38 views

Numerical Algorithm for $n \times n$ Matrix Inverse

I have to write a C program in which I have to compute the matrix inverse of a $n \times n$ matrix. Is there a convenient iterative process that I can use to do that? All I see is the co factor method ...
0
votes
0answers
18 views

Finding Inverse of a matrix using elementary transformations

So I have to find the Inverse of A. $$ A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 3 & 4 \\ 3 & 4 & 3 \\ \end{bmatrix} $$ By using elementary row or column transformations.. The ...
1
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1answer
16 views

Special Case Linear Solvers

I, and friends of mine, are interested in matrices which can be inverted / solved easily (i.e. in less than O(n^3)). I started to put together a github page dedicated to it and so far have identified: ...
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0answers
30 views

Do $p=2617$ and $q=3571$ have modular multiplicative inverse with $e = 17$?

I need multiplicative inverse of $17 \mod \left(\phi(p) \cdot \phi(q)\right)$. They are both prime, the totient of the product is $2616 \cdot 3570 = 9339120$. But $17$ is a factor of $9339120$, ...
1
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0answers
38 views

Any suggestions on how to find the inverse of this function?

What is the inverse of $$ d (X) = \frac {C \left( 1 - \frac b C \right)^2} {2 \left( 1-X \frac C b \right)} + \frac {X^2} {2 v (1-X)} - 0.65 \sqrt[3]{ \frac c {v^2}} X^{2+ \frac b C}$$ where $b, C, ...
0
votes
0answers
10 views

Invariance of Frobenious norm under transformation.

Can we say for every invertible square matrix $\mathbf{P}$, $\Vert\mathbf{X-B}\Vert_F^2=\Vert\mathbf{P^{-1}(X-B)}\Vert_F^2$. And does this hold true for non-square matrix $\mathbf{P}$ under some ...
0
votes
4answers
72 views

How does $1 + \tan^2x = 1/\cos^2x$?

I am unable to see why $$1 + \tan^2 x= 1/\cos^2x$$ I have looked into the topic anad I am familiar with the reciprocal ratios of cosec, sec, and cot. but cannot derive how this statement makes sense. ...
0
votes
1answer
17 views

Having a holomorphic function $h$ that's the inverse of a function $f$, it's also the inverse for a continuation of $f$

Let $\gamma: [0, 1] \to \mathbb{C}$ be a (continuous) path, $\gamma(0) \in D$, $(f, D)$ a tuple of a holomorphic function $f: D \to \mathbb{C}, D \subseteq \mathbb{C}$ a simply connected open set. Let ...
2
votes
1answer
48 views

Inverse of matrix with particular structure

I have a square invertible matrix $A=[c, c^2, c^3 \dots c^n]$ where $c \in \Bbb R^n$. Are there any known fast tricks for inverting it? Edit: $c$ is a column vector and raising it to a power is to ...
6
votes
3answers
223 views

Multiplicative inverse of $0$

If I'm not mistaken, in a ring with identity, the additive identity cannot have a multiplicative inverse. I'm trying to prove this. Here's my attempt so far: Suppose $0\cdot a=1$ $$0\cdot a=1$$ ...
0
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0answers
37 views

Using Chinese Remainder Theorem to find an integer $x$ for which $ x\equiv 3\pmod 4 x\equiv 5\pmod 9 x\equiv 10\pmod {35} $

Hello I have got problems with understanding the reduction method in CRT. We have got system like this $$x\equiv 3\pmod 4$$ $$x\equiv 5\pmod 9$$ $$x\equiv 10\pmod {35}$$ There is a way to do this ...
0
votes
0answers
31 views

What approximations for the Gamma function's inverse appear to work 'best'?

So I was wondering how we approximate the inverse of the Gamma function, where I tried a few methods: Lagrange inversion theorem: $$\Gamma^{-1}(z)=a+\sum_{n=1}^{\infty}\lim_{w\to ...
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1answer
27 views

Inverse trigonometric expansion related question

I know expansions for $\sin^{-1}(x)+\sin^{-1}(y)$, but does there exists any expansion for $\sin^{-1}(x \pm y)$ if not then what is the reason?
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2answers
61 views

How do I prove that $(ABC)^{-1} = C^{-1} B^{-1} A^{-1}$ [closed]

Please help me answering this problem! thank you :) Prove that for any nonsingular matrices $A$, $B$, and $C$, the equation $$(ABC)^{-1} = C^{-1}B^{-1}A^{-1}$$ holds. (Hint: Assume $D$ is the ...
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vote
4answers
14k views

Product of inverse matrices $ (AB)^{-1}$

I am unsure how to go about doing this inverse product problem: The question says to find the value of each matrix expression where A and B are the invertible 3 x 3 matrices such that $$A^{-1} = ...
0
votes
1answer
12 views

Relation of numerical stability of matrix inversion and it's determinant

I have been taught that "inverting a square matrix with small determinant is numerically unstable because it is close to singular"? Is this right opinion?