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 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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(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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51 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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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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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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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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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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42 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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165 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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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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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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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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29 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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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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19 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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47 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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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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28 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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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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Why Gaussian elimination can be used for matrix inversion?

Matrix inversion can be found by Guassian elimination, but what puzzle me is why does that method works. If possible, can you give me a proof about this method. Thank you. For your information: [B] ...
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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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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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21 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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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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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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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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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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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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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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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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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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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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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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58 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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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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29 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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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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41 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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49 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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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 ...
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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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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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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 ...
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1answer
23 views

Complex Field - Proving $\left(\frac{z_1}{z_2}\right)^{\star} = \left(\frac{z_1^{\star}}{z_2^{\star}}\right)$

Like the title states, I'm trying to prove that $\left(\frac{z_1}{z_2}\right)^{\star} = \left(\frac{z_1^{\star}}{z_2^{\star}}\right)$ where z is a complex number and z* is its conjugate. I keep ...
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Is there a closed form for the inverse of $y=x^{x^x}$?

It's pretty well known, and easy to derive, that $y=x^x$ has the inverse $y=\frac{\ln x}{W(\ln x)}$. I've had no luck trying to work out the inverse of any larger power towers, though. Is there any ...
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1answer
39 views

Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible for $A$ invertible and $B$ non-zero matrix

Let $A$ and $B$ be $n×n$ real square matrices. Matrix $A$ is an invertible and $B$ is a non-zero matrix. a)Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible b) Let $B= uv^T$ for $u,v \in \Bbb ...
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202 views

How do I solve $x^5 +x^3+x = y$ for $x$?

I understand how to solve quadratics, but I do not know how to approach this question. Could anyone show me a step by step solution expression $x$ in terms of $y$? The explicit question out of the ...
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1answer
83 views

How to find $10^{-1}$ in $\{0, 2, 4, 6, 8, 10, 12\} \subseteq \mathbb Z_{14}$?

Here is my question: Consider the subset $S = \{0, 2, 4, 6, 8, 10, 12\}$ in $\mathbb Z_{14}$, with the operations of addition and multiplication in $\mathbb Z_{14}$. (a) Show that $S$ has ...
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128 views

Determinants of Matrices det(4A) equals?

Suppose A is a 4 x 4 matrix such that det(A) = 1/64. What will det(4A^-1)^T be equal to? Here's my thinking, det(A^T) = det(A) I has no effect on the determinant. And det(A^-1) = 1/det(A) so ...
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68 views

Derivative of the Inverse Cumulative Distribution Function for the Standard Normal Distribution

As the title says, I am trying to find the derivative of the inverse cumulative distribution function for the standard normal distribution. I have this figured out for one particular case, but there ...