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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Can we find the inverse for a vector

Can we inverse a vector like we do with matrices, and why ? I didn't see in any linear algebra course such a concept of vector inverse and I was wondering if there is any such thing and if not, why.
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1answer
37 views

Inverse of $3$ by $3$ matrix with non-constant entries.

I'm solving a question in nonhomogenous ordinary differential equation system $x'=Px+q$, and to solve my question I need to compute the inverse of the matrix $A=\begin{pmatrix}e^{-2t} & e^{-t} ...
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1answer
29 views

How to find the inverse of the matrix over $\mathbb Z_5$

How to find the inverse of the matrix over $\mathbb Z_5$ $$ \left( \begin{matrix} 1 & 2& 0\\ 0 &2& 4 \\ 0& 0& 3\\ \end {matrix} \right) $$
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1answer
20 views

Next step to show that these matrice expressions are equal?

This is a problem from Discrete Mathematics and its Applications I know invertible means it is possible to take the inverse of this matrix. This is definition of a power of a square matrix from my ...
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1answer
25 views

Is the inverse of a causal function causal?

I am wondering if the inverse of a causal function is causal. I'll illustrate what I mean with an example: Assume $f$ is a bijection of $\mathbb R^2$ in $\mathbb R^2$. I assume $f$ is causal in the ...
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0answers
20 views

Inverse function theorem and Implicit function theorem.

I have been trying to prove that implicit function theorem implies the inverse function theorem. Be $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $\det[DF(x_0)]\neq 0$ for $x_0 \in ...
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2answers
16 views

Order of Inverse Operations

so this is a very simple question but I am having a tough time with it. So it's finals week and I'm studying up for an Algebra 2 final. The only part I am having trouble with is finding the inverse ...
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3answers
85 views

Find all matrices where the matrix is its own inverse and the determinant is 1

I need to find all the matrices: $$\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$ such that $$ad-bc=1$$ and $$A^{-1}=A$$ How would I go about doing this? I know that $AA=I^2$, ...
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1answer
11 views

Suppose that $p(x)=1/4x^4−2/3x^3-5/2x^2+6x-1/12 $withDom(p)=[1,2].Define$q(x)=p^−1(x)$. Show, algebraically, why q(x) exists

I don't know where to start. What does it means to define $q(x) = p^-1(x)$?
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1answer
39 views

Let $f(x) = \exp (x^2 − x + 6)$. Choose Dom(f) so that $f^{−1}$ exists. What is $f^{−1}$ and Dom($f^{−1}$) in your case?

I have already got $$y=\exp(x^2-x+16)$$ $$\ln y = x^2-x+6$$ $$\ln x=y^2-y+6$$ I know for getting inverse function we need to solve for $x$, but what should i do in this case?
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3answers
51 views

Invert a $2\times 2$ Matrix containing trig functions [duplicate]

Invert the $2\times 2$ matrix: \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} My thought was to append the $2\times 2$ identity matrix to the right ...
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1answer
37 views

Relation between $\tan^{-1}(x)$ and $\cot^{-1} (x)$

Suppose we've got $$I_1=\int_{-1}^{1} \tan^{-1}(x) + \tan^{-1} \left(\frac{1}{x}\right)$$ and $$ I_2=\int_{-1}^{1} \cot^{-1}(x) + \cot^{-1}\left(\frac{1}{x}\right)$$ So how can we relate $I_1$ and ...
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1answer
11 views

inversion of a symmetric matrix after that a column has been changed

Suppose $Z\in \mathbb R^{n\times k}$ and $S=Z^TZ$. Let now $Z(i, x)$ be the matrix $Z$ where the $i-th$ column has been replaced with $x$. Given $S^{-1}$ is there a quick way to compute ...
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1answer
13 views

Invertibility of $X^TX$ when sever multicollinearity in regression

I am studying about multicollinearity in regression and in the book it says, "if there is severe (but not perfect) multicollinearity, two or more predictor variables are highly correlated, so $X^TX$ ...
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3answers
122 views

Existence of continuous angle function $\theta:S^1\to\mathbb{R}$

Let $S^1\subseteq\mathbb{C}$ be the unit circle and let $U\subseteq S^1$ be open. How to show that there exist a continuous function $$\theta:U\to\mathbb{R}$$ such that $$e^{i\theta(z)}=z$$ for all ...
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4answers
55 views

Given A is a nil-potent matrix (given $ A^k=0 $), prove that A-I is invertible. Is my proof correct?

Given $A$ is a nil-potent matrix (given $A^k=0$), prove that A-I is invertible. I have proved the statement using contradiction, and I want to know if it is correct: Let $ A-I \neq I.$ ...
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2answers
53 views

Inverse of a unipotent matrix

Show that all unipotent matrices are invertible. Also, specify a formula for the inverse of a unipotent matrix. Now, I've tried to approach the problem using the determinant: a matrix is unipotent, ...
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1answer
27 views

Invertible “Sigmoid + x” function

I need an invertible function that represents a smooth transition between two straight, parallel line segments, like this: Depicted is $f(x) = -0.3/(1+e^{-10*(x-p)})+0.3/2+x$ (where $p$ is the ...
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1answer
26 views

Would there be no input or input does not exist?

This problem is from Discrete Mathematics and Its Applications. And the definition of inverse from the book: For part 43 (c), would the inverse not exist? For the floor function, in terms of $f(a) ...
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1answer
20 views

inverse of a power series with one specific solution

I have a school assignment and for now, I don't know where to start, I have to show that there exist a surrounding $U$ of $0$ where the following is true: If $y\in U$ , the equation $y=\frac{x}{f(x)}$ ...
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1answer
27 views

Matrix Inversion acceptable Condition Numbers

When considering matrix inversion it is worth while worrying about the condition number of the matrix you wish to invert. Matrices that are poorly conditioned can often create inaccurate results. This ...
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2answers
66 views

An element $a$ of a monoid $M$ is invertible iff there exists $x\in M$ such that $axa=1$

An element $a$ of a monoid $M$ is invertible iff there exists $x\in M$ such that $axa=1$ I can't do this one. How do I get started? It looks like it is saying there is only an inverse if ...
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1answer
80 views

Finding inverse of $f(x) =\frac{\ln(x)}{x}$

How do you find the inverse of the following function $$f(x) = \frac{\ln (x)}{x}$$ What looked like a simple question made my head hurt during exam.
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5answers
81 views

Why is the left inverse of a matrix equal to the right inverse? [duplicate]

Given a square matrix $A$ that has full row rank we know that the matrix is invertible. So there is a matrix $B$ such that $$ AB=1 $$ writing this in component notation, $$ A_{ij}B_{jk}=\delta_{ik} ...
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2answers
127 views

Given a finite metric space, are the matrices of triangle inequality errors invertible?

I have been working on some problems regarding finite metric spaces and have already proven/positively answered the following statement/question if the underlying metric has additional properties. Now ...
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0answers
22 views

complex and decimal tetration

So recently in the blog post on tetration, it talked about tetration with nice clean powers (calling them these because I don't know the right term). But how does it work when given a complex power? ...
2
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1answer
34 views

Proving this function is an open map

Prove the function $f(x, y, z) = (x^3, y^2-z^2, yz)$ is an open map from $\mathbb{R^3}$ to $\mathbb{R^3}$ (i.e for every open set $U$ of $\mathbb{R^3}$, $f(U)$ is open). I know, as an application of ...
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1answer
32 views

Solving equations with matrices

Say I have $4$ simultaneous equations \begin{cases} 4.3S_1 - P = T \\ 8S_2 - P = T \\ 5.5S_3 - P = T \\ S_1 + S_2 + S_3 = T. \end{cases} I'm trying to solve these in Excel using MINVERSE and MMULT ...
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1answer
70 views

If $f(AB) =f(A)f(B)$, then $A$ is inversible iff $f(A)\neq 0$

Let $f:\mathscr M_n(\mathbb K) \to \mathbb K$ be a non constant function such as $f(AB) = f(A)f(B)$ for all $A,B$ in $\mathscr M_n(\mathbb K)$. The question is to show that $M\in GL_n(\mathbb K)$ iff ...
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0answers
20 views

Inverse transversal is perfect if and only if $li=i^0lil^0$ for $i \in{I}$, $l \in{\Lambda}$

I am attempting show that an inverse transversal is perfect if and only if $li=i^0lil^0$ for $i \in{I}$, $l \in{\Lambda}$. An inverse transversal is perfect if $li(li)^0=(li)^0li$. I have shown the ...
3
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2answers
187 views

find x where $x^{11} \mod 41 = 10$

In a previous part of the question, I am asked to find $11^{-1} \mod 40$. I've done that, the answer's $11$. The question continues: find $x$ where $x^{11} \mod 41 = 10$ showing how you could get ...
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3answers
55 views

Solve logarithmic equation for $x$ to find the inverse of $f(x)= \ln(x+\sqrt{x^2+1})$

Let $f(x)= \ln(x+\sqrt{x^2+1})$. Find $f^{-1}(x)$. Here is what I got so far: $y= \ln(x+\sqrt{x^2+1})$, rewrite as $x= \ln(y+\sqrt{y^2+1})$, then $$e^x= y+\sqrt{y^2+1}$$ $$e^x-y= \sqrt{y^2+1}$$ ...
3
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1answer
71 views

Questions about matrix rank, trace, and invertibility,

(a) Prove that a square matrix $T$ of rank one has $\text{tr}(T)=0$ if and only if $T^2=0$. (b) Consider a matrix $A$ of the form $A=aI+T$, where $a\ne0$, $I$ is the identity matrix, and $T$ has ...
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1answer
80 views

Mistake in integrating the inverse of a function?

The problem I have is: $$\int_0^{\frac1{\sqrt{3}}}\sqrt{x+\sqrt{x^2+1}}dx$$ I'm not asking how to solve it, I'm asking if anyone can point out to me the error that I have made: By differentiating ...
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0answers
63 views

Fastest way to find modular multiplicative inverse

I am looking for a fast way to find the modular multiplicate inverse of an integer $a$ in mod $p$. I am mainly interested in ...
4
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1answer
48 views

inverse of a tridiagonal matrix

Let $${A_{n \times n}} = \left[ {\begin{array}{*{20}{c}} {-2}&{1}&{}&{}&{}\\ {1}&{-2}&{1}&{}&{}\\ {}&{1}&{\ddots}&{\ddots}&{}\\ ...
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2answers
60 views

Finding inverse of a composite function

Let $f (x) = x^{3}+x$ and $g (x) =x^{3} -x$ for all x. I have to find derivative of $g\circ f^{-1}$ at $x=2$. My textbook did this: $(g \circ f^{-1})' (2) = \lim \limits_{h \to 0} \dfrac{g \circ ...
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0answers
10 views

Efficient inverses of many related matrices [duplicate]

Say I have a $N$-by-$N$ positive definite real matrix $\Sigma$ and I wish to compute the inverses (or equivalently Cholesky decompositions) of $(\Sigma + a_k I)^{-1}$ for a set of $K$ positive $a_k$. ...
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1answer
39 views

Finishing a proof: $f$ is injective if and only if it has a left inverse

I've already done a lot of searching (in particular: https://www.proofwiki.org/wiki/Injection_iff_Left_Inverse) to try to prove this statement: $f: A \to B$ is injective if and only if it has a ...
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2answers
58 views

Is the preimage of a bounded set also bounded?

I need to prove the following statement: Let $f:\mathbb{C}\rightarrow\mathbb{C}$ a continuous function and $B \subseteq \mathbb{C}$ bounded, implies, that the set $A=f^{-1}(B)$ to be bounded. I do ...
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1answer
36 views

Finding a differentiable inverse of $f(x)=x/\cos x$

Let $$ f:\left(-\frac{\pi}{2},\frac{\pi}{2}\right) \rightarrow \mathbb{R} $$ be defined by $$ f(x) = \frac{x}{\cos x}. $$ We're supposed to show that $f$ has a differentiable inverse $$f^{(-1)}$$ ...
0
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2answers
46 views

What's $\int \frac{1}{\sqrt{25-x^2}}$ [duplicate]

What is $$\int \frac{1}{\sqrt{25-x^2}}$$ WolframAlpha says $\sin^{-1}(\frac{x}{5})$ while I got $\frac{1}{5}\sin^{-1}(\frac{x}{5})$. What is correct? Thanks in advance.
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2answers
51 views

Prove that a continuous inverse-transformation of $f: [0,1) \cup \{ 2 \} \to [0,1]$ exists

I am having this transformation $f: [0,1) \cup \{ 2 \} \to [0,1]$ $$f(x) = \begin{cases} x & x \neq 2 \\1 & x = 2 \end{cases}$$ I've already proved that it is continuous. Question: Is ...
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1answer
27 views

Compute $\left(a_{i}A+B\right)^{-1},\qquad i=1,\ldots,N$ efficiently?

I need to compute the inverse matrix: $$(a_i A+B)^{-1}, \qquad i=1,\ldots,N$$ where $N$ is a large number. $A$ and $B$ are general $M\times M$ matrices independent of $i$. The only thing that ...
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0answers
20 views

Inverse transformation of continous transformation is bounded

I am having a continous transformation: $f: \mathbb C \to \mathbb C $ with $B \subseteq \mathbb C $ bounded. Now I want to proove that $ A = f^{-1} (B)$ is bounded! How can I proove that this ...
8
votes
2answers
334 views

Why aren't integration and differentiation inverses of each other?

Integration is supposed to be the inverse of differentiation, but the integral of the derivative is not equal to the derivative of the integral: $$\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\int ...
4
votes
1answer
23 views

inverse of a point $p$ respect to the circle $|z-z_0 |= r$ in complex

I was solving a problem to find the inverse of a point $p$ respect to the circle $|z-z_0|=r$. In my question I had to find inverse of $1+i$ w.r.t circle $|z+1-2i| = 2$. I applied the formula $q = z_0 ...
0
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1answer
50 views

Derivative of scalar function with respect to vector

Suppose I have three constant symmetric matrix $\mathbf{M}_{n\times n}$, $\mathbf{C}_{n\times n}$ and $\mathbf{D}_{n\times n}$ and two variable vectors $\mathbf{q}_{n\times 1}$ and ...
1
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1answer
33 views

Finding the inverse of a recursive function?

Let's say I have this function $$f(x) = \sum_{i=0}^{x-1}f(i)$$ provided $f(0) = 0, f(1) = 1$ and $x \in \mathbb Z$. This function is evidently one-to-one on $[3, \infty) $. Is there an inverse to this ...
0
votes
1answer
51 views

The invertibility of matrix $(I - XX')$?

$I$ is an identity matrix of size $n \times n$. $X$ is a matrix of size $n \times k$(Assuming $k \leq n$). As we know, $(I+XX')$ is invertible. Because $(I+XX') = (I(blank)X)*(I(blank)X)'$, where $(I ...