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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Finding only first row in a matrix inverse

Let's say I have a somewhat large matrix $M$ and I need to find its inverse $M^{-1}$, but I only care about the first row in that inverse, what's the best algorithm to use to calculate just this row? ...
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27 views

Inverse of a function containing the ceiling function over the natural numbers

I am wondering if there exists an inverse function for $\lceil{e^{x}}\rceil$ over the natural numbers. I don't think it is a trivial task to derive an inverse function for a function containing a ...
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38 views

How to find the inverse function of Euler's number?

Given: $f(x)= \dfrac{e^x}{1+9e^x}$ , what steps would I take to find its inverse? I tried following the steps on finding the inverse of a normal function but I keep getting one of the variables to ...
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42 views

What is the inverse kernel to this integral transform

What is the associated inverse kernel to the integral transform $T$ defined by \begin{align*} (Tf)(u) & = \int_{-\infty}^{0} \hat{f}(s)\exp((2i\pi+c)us)\ ds + \int_{0}^{+\infty} ...
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Question concerning Preimage

Let $f$ be the map from $\mathbb{R} \to \{a,b,c\}$ defined by \begin{equation} f(x)=\begin{cases} a &\text{if} \quad x>0 \\ b & \text{if} \quad x<0 \\ c &\text{if} \quad x=0 ...
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46 views

How to show that $AX=B$ has unique solution for invertible matrix $A$

If $A$ is an invertible $n \times n$ matrix, show that $AX=B$ has a unique solution for any $n \times k$ matrix $B$. I'm not sure where to start. What I have is that, if $A$ is invertible then ...
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1answer
53 views

Integration of a function containing inverse trigonometric functions

Q. $$\int \sin\left\{2\tan ^{-1}\left(\sqrt{\frac{3-x}{3+x}}\right)\right\}dx$$ $\implies$ $$\int \sin\left\{\sin ...
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25 views

Show that sum of elements of rows / columns of a matrix is equal to reciprocal of sum of elements of rows/colums of its inverse matrix

Suppose $A=(a_{ij})_{n\times n}$ be a non singular matrix. Suppose sum of elements of each row is $k\neq 0$, then the sum of elements of rows of $A^{-1}$ is $\frac{1}{k}$. Let ...
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1answer
19 views

Finding Inverse of exponential function

$f(x)=\frac {e^{(x)}} {(1+2e^{(x)})}$ I'm having trouble finding the inverse of this function algebraically.
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40 views

(n x n) Matrix multiplying itself with its inverse to form the (n x n) identity matrix

Is it ok to say Matrix A, with it's inverse, form the Identity Matrix? Thanks
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52 views

Prove that $g \circ f$ is a one-to-one function

Let $f$ and $g$ be one-to-one functions such that the domain of $f$ is $A$, the range of $f$ is $B$, the domain of of $g$ is $B$, and the range of $g$ is $C$. Prove that $g \circ f$ is a one-to-one ...
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170 views

Proof of Matrix Norm (Inverse Matrix)

Show for any induced matrix norm and nonsingular matrix A that $$ \left\|A^{-1}\right\| ≥ (\left\|A\right\|)^{-1} $$ where $$ \left\|A^{-1}\right\| = ...
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1answer
33 views

Inverse matrices properties.

I know about the properties of matrix multiplication for multiplication such as $A(BC)=(AB)C$. However I'm curious if $(AC)B$ would also have the same value. I'm asked to represent $A$ in terms of $B$ ...
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3answers
64 views

Find the inverse with respect to the binary operation $a ∗ b = a + b + a^2 b^2$

A binary operation on $\mathbb{R}$: $a * b = a + b + a^2 b^2$ The neutral element I found to be $0$. Then I need to find an invertible element having two distinct inverses. I don't know where to ...
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40 views

If there is a mapping of $B$ onto $A$, then $2^{|A|} \leq 2^{|B|}$

If there is a mapping of $B$ onto $A$, then $2^{|A|} \leq 2^{|B|}$. [Hint: Given $g$ mapping $B$ onto $A$, let $f(X)=g^{-1}(X)$ for all $X \subseteq A$] I follow the hint and obtain the function $f$. ...
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1answer
43 views

Following flash, a camera's battery begins to recharge the flash’s capacitor, which stores electric charge given by $Q(t) = Q_0(1 − e^{−t/a})$ [closed]

(The maximum charge capacity is $Q_0$ and $t$ is measured in seconds). (a) Find the inverse of this function and explain its meaning. (b) How long does it take to recharge the capacitor to 90% of ...
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1answer
171 views

Linear algebra proof regarding matrices

I'd like a hint rather than a full solution. The problem I am considering is the following: $X$ is an $n\times m$ matrix $Y$ is $m\times n$ Show that $(I - XY)^{-1}\cdot X = X\cdot(I - ...
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22 views

Slight help with inverse trigonometry question

I apologize for the lack of LaTeX, i will try to learn LaTeX and update this question as soon as possible. I am having some trouble with an inverse trigonometry question and was hoping that someone ...
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2answers
42 views

for $k\neq 0, -1, 1$, find the inverse of the matrix

for $k\neq 0, -1, 1$, find the inverse of the matrix $$\begin{pmatrix} k&0&0\\ 1&k&1\\ -1&1&k \end{pmatrix}$$ how am I supposed to solve this? all I can think of is plugging ...
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29 views

Study the associative and commutative properties and neutral and inverse elements of these groups

Group m*n = max(m,n) on Z and N So i showed its associative by m,n,p in Z and (m*n)*p = max(m,n)p =max(m,n,p) And m(n*p) = m*max(n,p) = max(m,n,p) Commutative m*n = max(m,n) and n*m = max(n,m). I ...
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1answer
35 views

Field Proofs with Multiplicative Inverses

I've been staring at these two for a while and I can't come up with anything concrete to start. Hints on how to begin would be greatly appreciated, full solutions are not necessary (and preferably ...
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1answer
47 views

expansion of matrix inverse

I would like to invert a square matrix $L$. One can write it as a sum of two matrices, one containing the diagonal terms ($D$) and the other the off-diagonal ones ($A$). $$L = D+A$$ I would like ...
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1answer
20 views

transpose and inverse multiplication

Given: $$A_{(n,n)} , B_{(n,n)}$$ A and B are invertible, is it possible that : $$(A^t B^t)^{-1} A^{-1} B^{-1} = I$$ I guess no, should this be true only if the AB=BA= orthogonal matrix ?
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50 views

Closed form of the inverse of a function

Does anyone know what the analytic form of the inverse of $f(x)=e^x+x$? Thanks in advance
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1answer
35 views

How to get the inverse function of this one?

Let's have function $$ \psi (x) = -\frac{1}{ax} - \frac{b}{a^2}\ln(x) + \text{const} + O(x). $$ I have read that the inverse function is written in a form $$ \psi^{-1}(t) = -\frac{1}{at} - ...
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29 views

Arc-Gamma Function.

Is there an arc-gamma function? Where gamma(x) = y... Arc-gamma(y) = x. I've searched and found something called DiGamma Function, but when I substituted it didn't seem to be "arc" but something ...
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3answers
93 views

What is the best way to find $g(x) = f^{-1}(x)$?

so the problem I have is if $f(x) = \sqrt{x+3} - 2$ and it asks to find the solution of $f(x) = f^{-1}(x)$. So i know to find the inverse, which I got as $f^{-1}(x) = (x+2)^2-3$. So to find the ...
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2answers
38 views

Getting inverse of polynoms with trigonometric functions

I'm trying to get the inverse of $$f(x) = \cos(x) + 3x$$ I tried it by definition of $\cos(x)$ with no luck: $$\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}+...$$
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66 views

Showing that a matrix is invertible and finding its inverse

I'm incredibly rusty at linear algebra, and in preparation for my course I've been doing some review questions. I've been staring at this one for a half hour and still don't know how to approach it: ...
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1answer
30 views

Case Deletion Diagnostics

I have NO idea how to approach this problem. I don't see any connection between the corollary and the formula we need to prove. Does anyone have any hints? Corrolary: If $\mathbf{A}$ and ...
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22 views

Calculating the left pseudoinverse of a Matrix whose columns are Probablity Mass Functions

I have a matrix $A_{m\times n}$, where $A_j$ , a column of $A$ represents a probability mass function, and so the sum over the column is 1. This is true for all the columns of A, i.e. $\forall j \in ...
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3answers
33 views

Simple inverse function of $\frac{1-2x}{1+x}$

Just started learning about inverse functions, and got stuck on this one: $$f(x) = \frac{1-2x}{1+x}$$ So I tried multiplying by $(1+x)$ on both sides and got $y+yx = 1-2x$ but that doesn't seem to ...
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1answer
26 views

How to find the inverse of a function involving e with a coefficient?

I was wondering how I would find the inverse of the following function, since the e has a co-efficient: $\frac{e^x}{1+2e^x}=y$ I got as far as $\ln y+\ln(2e^x) = \ln e^x$, which would be changed ...
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0answers
42 views

Invertibility of an operator involving inner product

Let $H$ be a Hilbert space with basis $b_i$. For all $t$, let $f(t;\cdot,\cdot)$ be an inner product on $H$. For each $j$, is $$\int_0^T \sum_{i=1}^\infty f(t,b_i,b_j)x_j(t)=0$$ uniquely solvable for ...
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83 views

Given $f(x)=\int_5^x \sqrt{1+t^2}\,dt$, find $(f^{-1})'(0)$

If $f(x)=\int_5^x \sqrt{1+t^2}\,dt$, find $(f^{-1})'(0)$. Here is what I have done so far. I have took $f'(x)=(1+x^2)^{1/2}$ and I have found $1/f'(0)$ which should equal $1$. I don't think this ...
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1answer
26 views

Laplace transform, Inverse Laplace transform

Let $(\mathcal{L}f)(s)$ be the Laplace transform of a piecewise continuous function $f(t)$ defined for $t\geq 0$. If $(\mathcal{L}f)(s)\geq 0$ for all $s\in\mathbb{R^+}$ does this imply that $f(t)\geq ...
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1answer
101 views

Finding that values k that make this matrix invertible without using the determinant

The matrix in question is A = [(1,1,1),(1,2,k),(1,4,k^2)]. I know that I can row reduce the matrix to rref, which should in theory leave me with some k values in the matrix from which I can see what ...
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0answers
20 views

Hitting time and its distribution

÷I'm reading an italian book about casual process (Probabilità e modelli aleatori of Enzo Orsingher). At pag 105 there's the probability of the stopping time $T_\beta$. $$P\{T_\beta \leq ...
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1answer
33 views

inverse laplace transform of $s/(s^2+6s+13)$

Hi can anyone help with this inverse Laplace transform $$s/(s^2+6s+13) $$ I tried to do partial fraction $s+3/(s+3)^2+4 - 2/(s+3)^2+4$, but then I don't know what to do next...
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1answer
33 views

Do lines between determinants pass through the inverse?

Let A be a $2 \times 2$ matrix whose inverse also exists. If I was to draw a line from each of the 3 vertices (that are not the origin) of the determinant of A, to the 3 vertices of the determinant of ...
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1answer
63 views

To find the inverse of an implicit function

I have a function $t(f)$ here: $t(f) = T(sin(2\pi f/B)/2\pi + f/B) $ for $[-B/2 \le f \le B/2]$. $B$ and $T$ are constants. How to find the inverse of this function that is $f(t)$ using numerical ...
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2answers
57 views

If $f$ takes $[-1,1]$ onto $[-1,1]$ then $f^{-1}(\{f(0)\})=\{0\}$

Consider the statement: If $f$ takes $[-1,1]$ onto $[-1,1]$ then $f^{-1}(\{f(0)\})=\{0\}$. My book tells me this is suppose to be false, but I don't understand why. We know: If $f:X\to Y$ has ...
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0answers
32 views

Finding inverse of a general linear transform

I'm not a mathematician, so I may abuse some notation here. Please comment for any clarification. Let's define a general linear transform as $$\int_XK(\mathbf{\omega},x)f(x)dx$$ where $X$ is some ...
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1answer
27 views

On matrices sharing the same smallest nonzero eigenvalue and related eigenvector

Suppose that $A$ and $B$ are square matrices with the proper size, then what kind of condition does $B$ have to satisfy such that $A$ and $AB$ share the same smallest(largest) nonzero eigenvalue and ...
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43 views

Canceling a Limit with 'Inverse Limit'

Is there an operator that cancels a limit (inverse limit)? For example $lim_{x\rightarrow0}e^{x}$: Can one take an 'inverse limit' to cancel with this limit and output $e^{x}$? Let A='inverse limit', ...
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3answers
50 views

What would be the inverse function for the following condition?

What would be the inverse function condition for the above question.
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0answers
40 views

Determining $f^{-1}(3)$ without knowing $f^{-1}(x)$ but given $f(1)=3$ and $f'(x)>0$.

I have a continuous function $f(x)$ and I want to find $f^{-1}(3)$, but I can't find $f^{-1}$ directly. I know that $f(1)=3$ and $f'(x)>0$ for all x. Because the function is continuous and always ...
3
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0answers
64 views

Inverse of $x^2+\log^2\cos x$

I'm looking for the inverse of $$f(x)=x^2+(\log\cos x)^2$$ Where $f$ is defined from $[0,\pi/2)$ It dosen't have to be closed form, a sum, an integral or some special functions would be of interest ...
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2answers
23 views

Prove that $B$ is invertible:$B = A_{11} - A_{12}A_{22}^{-1}A_{21}$ if…

Let $$A = \begin{bmatrix}A_{11}&A_{12} \\ A_{21}&A_{22} \\ \end{bmatrix}$$ Prove that $B$ is invertible:$$B = A_{11} - A_{12}A_{22}^{-1}A_{21}$$ $$$$ in case of $A$ and $A_{22}$ being ...
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0answers
33 views

Formal inverse of a matrix ressembling Fourier's matrix

What is the formal inverse of a square $N\times N$ matrix $A$ with entries $A_{ij}=a^{(i-1)(j-1)}$? When $a$ is the $N$th root of unity (i.e. $a=\exp(2 \pi i/N)$), then $A$ is the Fourier matrix and ...