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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Inverse Laplace tranform via the table formulas

In my inverse Laplace table there is this inversion "formula": $(1) \frac{1}{s-a} \rightarrow e^{at}$ I understand that $\mathcal{L}^{-1}[\frac{1}{s+4}] = \frac{1}{2}\sin(2t)$ But why can I not do ...
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24 views

If $\sin^{-1}\frac{2a}{1+a^2}-\cos^{-1}\frac{1-b^2}{1+b^2}=\tan^{-1}\frac{2x}{1-x^2}$ then what is value of x?

If $\sin^{-1}\frac{2a}{1+a^2}-\cos^{-1}\frac{1-b^2}{1+b^2}=\tan^{-1}\frac{2x}{1-x^2}$ then what is value of x? Solution $\tan^{-1}x=\tan^{-1}a-\tan^{-1}b=\tan^{-1}\frac{a-b}{1+ab}$ ...
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51 views

Tan inverse summation

$$S=\sum\limits_{i=1}^{4}\tan^{-1} x_i$$ How to simplify this ? I think I will have to use this : but it looks too long a method . Is there a method or symmetrical way which yields ...
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41 views

Arithmetic modulo primes task

I'm dealing with a problem here. The problem is as follows: There is a set $Z_p=\{0,1,2,3,...,p-1\}$ where $p$ is a prime. From this set we form a new set $B=\{x+x^{-1}\mid x\in Z_p\}$, where the ...
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Closed form for elements of inverse matrix of lower triangular matrix of any size

If we have a lower triangular matrix $$A=\left(\begin{array}{rrrrr}a_{1,1}&0&0&\cdots&0\\a_{2,1}&a_{2,2}&0&\cdots&0\\a_{3,1} &1_{3,2}&a_{3,3}&\cdots&0\\ ...
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19 views

a matrix inverse problem

Given a matrix $X$, let $D$ be a diagonal matrix whose diagonal elements are row sums of $X$, let $I$ be an identity matrix. Now I have a resultant matrix of $Y=(I-X)^{-1}$, and I would like to ...
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1answer
21 views

Deriving an identity using the Woodbury matrix identity

I am working through an algorithm derivation in Kernel Adaptive Filtering: A Comprehensive Introduction by Liu, Principe and Haykin. The part I'm having trouble with is on page 104 if you have the ...
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1answer
12 views

Inversion of Boolean function Application

Asume you have a boolean function $f$ which takes $n$ parameters and gives $m$ results. In addition, you have a boolean function $g$ takes $p$ parameters and gives out $n$ results. You could ...
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48 views

Show that the inverse function to $f(x)=\int_{1}^{x}\frac{dt}{t}$ is differentiable

Show that the inverse function to $$f(x)=\int_{1}^{x}\frac{dt}{t}$$ is differentiable. I know that the integral is $\ln(x)$, but I don't know how to show that it is differentiable in a good way ...
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1answer
42 views

Cholesky, Inverse, and Determinant when updating the diagonal of a symmetric positive definite matrix

Suppose that $A$ is a symmetric positive definite matrix and assume its dimension $n$ is large. Let $I$ be the $n \times n$ identity matrix and $m \neq 0$ be a scalar. I'm interested in computing as ...
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51 views

Word problem about finding the inverse derivative

I have the following word problem. I need to find and interpret the meaning of the inverse derivative of a function. At a gas station, the function f(p) is the number of gallons of gasoline sold when ...
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4answers
58 views

Calculate inverse of matrix

If $$A=\begin{bmatrix} -5 & 1 & 0 & 0\\ -19 & 4 & 0 & 0\\ 0 & 0 & 1 & 2\\ 0 & 0 & 3 & 5\\ \end{bmatrix}, $$ how do I calculate $A^{-1}$? Is there any ...
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1answer
23 views

Is it Possible to Develop an inverse function using the function it self

Is it Possible to Develop (taylor expansion) of an inverse function by knowing the function it self ? If Yes ,Can you illustrate with a simple function I know that we use the identity formula $$ ...
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43 views

Finding a Matrix from Determinants

I've stumbled upon this problem on my homework, and I have no clue how to do it, and haven't found any help online: If I'm understanding this correctly, then $det(M) = ad - cb + eh - gf$ ? What I ...
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3answers
130 views

Calculate inverse of arbitrarily sized, lower triangle matrix with a specific pattern.

I have a matrix of the following form: $$A=\begin{bmatrix} 2 & 0 & 0 & 0 \\-1 & 2 & 0 & 0\\ 0 & -1 & 2 & 0\\ 0 & 0 & -1 & 2 \end{bmatrix}$$ which, in ...
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39 views

Inverse of a special matrix

Is there easy (analytical) way to find the inverse of the following matrix, where $C$ is a vector? $$ \begin{bmatrix} 1 & C^\top \\ C & CC^\top \end{bmatrix} $$
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21 views

Derivative of inverse function where inverse is known only numerically.

I have the following polynomial function in $\mathbb{R}$: $$f(x,a)=ax^3+x$$ However, $x$ also depends on $a$, so we should rather write: $$f(x,a)=ax(a)^3+x(a)$$ Now I need derivative of $f$ and ...
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33 views

Inverse of split functions

I don't know how to find the inverse is of a function when is split. Example, $\Bbb R_+$ is the set of positive real numbers. $f : \Bbb R \to \Bbb R_+$ $$f(x) = \begin{cases} 2-x & \text{if } ...
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22 views

check a matrix for positive semidefiniteness in the general case?

Here is the problem that I am facing. I know that for a positive semidefinite matrix to exist, the condition below must be satisfied: $x^\intercal Y x \geq 0$ where $x$ is a non-zero vector, i.e. a ...
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1answer
45 views

What does it mean for a matrix to change basis?

I understand what it means for vectors, i.e. $$ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} ...
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50 views

How to calculate the inverse of sum of a Kronecker product and a diagonal matrix

I want to calculate the inverse of a matrix of the form $S = (A\otimes B+C)$, where $A$ and $B$ are symetric and invertible, $C$ is a diagonal matrix with positive elements. Basically if the ...
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1answer
30 views

jacobian matrix of $f^{-1}$ at $(1,0,2)$ given $f(x,y,z)$

If $f(x,y,z)=(sin(xyz),(x+x^2)*cos(y),y)$ and $f$ has a local inverse in the neighborhood of $(0,1,1)$, how do I find the jacobian matrix of this inverse at $(1,0,2)$? I know from definition that ...
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Find the slope of the tangent line to the graph of $f^{-1}$

Given function $f$, find the slope of the line tangent to the graph $f^{-1}$ at the point on the graph $f^{-1}$. $f(x)=\sqrt{5x}$; $(4,\frac{16}{5})$? Here is what I have thus far: $f'(x)= ...
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How to solve an Inverse differentiation problem

If f is a one-to-one function where $f(3)=2$ and $f'(3)=6$, what is the value of $(f^{-1})'(2)$? I am not even sure where to start with this question. I was hoping someone can help $f$ of $3 =2$ and ...
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implementing modular multiplicative inverse.

I wish to implement the clifford cocks algorithm using GMP. In the encryption part: $c_1=t_1+at_1^{-1}\bmod n$. Following $( a b \bmod n ) = ((a \bmod n) \cdot (b \bmod n ))\bmod n$, I took the ...
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31 views

Integral of reciprocal of a piecewise linear function

Let, e.g. $$ f(x) = \begin{cases} x,\quad x<1, \\ 1,\quad x\geq1, \end{cases} $$ a piecewise linear function. Does the following hold for $g(x) = 1/x$? $$ \begin{align} g(f(x)) ...
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what function fulfills these conditions? [duplicate]

So I know that if $f(x) = x^{-1}$, than $f(f(x)) = x$ but $f(x)$ is not necessarily $x$. So now, is there $g(x)$ such that $g(g(x)) \neq g(x) \neq x$ but $g(g(g(x))) = x$? If so what is it, else why ...
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Inverse of 3-by-3 matrix

Hi, so this question is taken straight from khan academy help exercises, i know how to do it dynamically meaning using the determinant and the adjugate how i was trying to do it using guass bla bla ...
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14 views

prove that $X$ is invertible if and only if $Y$ is invertible. if $(-1)^i(1+i)x_i^T=y_i$

$X=[x_1,x_2,...,x_n]$ and $Y =$ $y_1\\y_2\\...\\...\\...\\y_n$ where $x_i$ and $y_i$ are column and row matrices respectively. $X$ and $Y$ are both $n$ x $n$ matrices. if $$(-1)^i(1+i)x_i^T=y_i$$ ...
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39 views

Find fundamental matrix of a 2x2 matrix with rank 1

$$ x'(t) = \left[\begin{array}{cccc}0&1\\0&t\end{array}\right]x(t)$$ I am having trouble computing the fundamental matrix. I get: $$ x1(t) = x2(0)*exp(0.5t^2) $$ $$ x2(t) = x2(0)*exp(0.5t^2) ...
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32 views

Inverse Sine and cosine

$\arcsin(\cos(x))=1/2$ Find $x$. I got $-1/2$ or $2\pi-1/2$, but I don't know the correct answer. I tried graphing unit circle.
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Inverse Function Theorem [closed]

Use the inverse function theorem to show that if $f \colon A \to \mathbb{C}$ is analytic and the derivative of $f$ is never $0$, then $f$ maps open sets into open sets.
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A square matrix with the diagonal and antidiagonal elements different from zero. Looking for some already known property.

I am interested in the properties of a matrix with elements different from zero only on the main diagonal and antidiagonal, like this: $$ \begin{matrix} a & 0 & 0 & h \\ ...
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1answer
58 views

If we have a square matrix thats invertible, do its row and column space coincide?

If we have a square matrix thats invertible, do its row and column space coincide? Regarding an nxn invertible matrix: -The row space of the matrix is R^n -The column space of the matrix is R^n ...
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32 views

compute the inverse function

Assume $h(x)$ is an invertible function. Let $g(x)=2+8h(4x+1)$. Find the inverse of $g$ in terms of $h^{-1}$ So following the usual steps to get the inverse function, I rearranged to get ...
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63 views

Show a matrix is invertible [duplicate]

How to show that $$A=\begin{pmatrix}1233&2344&1324&3456\\ 2342&11233&1432&13256\\234132&32432&1234567&43254\\423412&42354&452356&13245\end{pmatrix}$$ ...
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4answers
60 views

Calculate the multiplicative inverse modulo a composite number

I want to calculate $ 8^{-1} \bmod 77 $ I can deduce $ 8^{-1} \bmod 77$ to $ 8^{59} \bmod 77 $ using Euler's Theorem. But how to move further now. Should i calculate $ 8^{59} $ and then divide ...
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1answer
64 views

Summation of $\tan^{-1}$ series

I am given $$S=\sum\limits_{n=1}^{23}\cot^{-1}\left(1+ \sum\limits_{k=1}^n 2k\right)$$ On expanding the sigma series becomes $$S= 23\cot^{-1}(3)+22\cot^{-1}(5) + \cdots + \cot^{-1}(47)$$ And in tan ...
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Find the inverse function of $y = g(x) = 6 x^3 + 7$: $g^{-1}(y) =?$

The question states, Find the inverse function of $y = g(x) = 6 x^3 + 7$, $g^{-1}(y) =?$ I have tried setting the equation to $y$ and then solving for $x.$ This resulted in the answer ...
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Proof: The identity matrix is invertible and the inverse of the identity is the identity

How can i show that: $II^{-1} = I = I^{-1}I$ (the identity matrix is invertible) for all cases. And then proof that: $I^{-1} = I$ (The inverse of the identity is the identity). I don't know how start ...
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Proof: The inverse of the inverse matrix is the matrix.

If $A$ is a square matrix such that it is not singular, then $(A^{-1})^{-1} = A$ How can I prove this property? I would appreciate it if somebody can help me.
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118 views

How to find the inverse cosine without a calculator

How to find the inverse of: $$\cos(c)=\frac{1}{3}$$ In other words, i'm trying to solve for c and without a calculator. If it's hard or not possible, then how would you go about solving inverses in ...
2
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1answer
29 views

Each of the following functions f is bijective. Describe its inverse.

QA,B: Each of the following functions f is bijective. Describe its inverse. A: $$f:\mathbb{R} \rightarrow (0,\infty); \text{ defined by } f(x)=e^x $$ For this function, I said the inverse is: ...
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3answers
67 views

Linear Algebra Matrices

Determining the values of a for which the Matrix A has an inverse ! A= \begin{pmatrix} 1 & a & 1 \\ 2 & a+2 & 1 \\ 1 & 2 & a \end{pmatrix} How i solved it: ...
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213 views

Find the numbers that have an inverse modulo 11

I am trying to understand the inverse of a modulo. I want to find the numbers in the range 1,2,3...11 modulo 11 that has an inverse. I am confused and I can't understand how to identify which ...
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1answer
33 views

Box-Muller Independence Proof by Change of Variables (Help finding the Inverses)

Let $X_1=\cos(2 \pi U_1)\sqrt{-2 \log(U_2)}$ and $X_2=\sin(2 \pi U_1)\sqrt{-2 \log(U_2)}$ wher $U_1$ and $U_2$ are iid uniform (0,1). Prove that $X_1$ and $X_2$ are independent N(0,1) random ...
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45 views

Inverse trignometry

The following function models the length $L$ of each day (in minutes) in Manila, $t$ days after the spring equinox, which is March 22. $$ L(t) = 52 \sin\left(\frac{2\pi t}{365}\right)+728$$ What is ...
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2answers
50 views

Show that there exists a differentiable function $f$ s.t. $(f(x))^5+f(x)+x=0$

Show that there exists a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ s.t. $(f(x))^5+f(x)+x=0$ for all $x \in \mathbb{R}$ I am meant to use the Inverse Function Theorem for ...
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2answers
28 views

Find $\frac{dy}{dt}$ for the given x-values.

A point moves along the curve of the given equation such that $\frac{dx}{dt}$ is 2 cm/s. Find $\frac{dy}{dt}$ for the given values of $x$. $$y= \frac{1}{1+x^2};$$ $$x=-2, x=0, x=2$$ I've just ...
0
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1answer
38 views

Find the inverse of defined operation $\Delta$

We defined the operation $ \Delta $ as $ (a,b) \Delta (c,d) = (ac + \delta bd, ad + bc) $ where $ a, b, c ,d \in \mathbb{Q} $ I have already proven that this operation is both commutative and ...