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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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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35 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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3answers
35 views

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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2answers
59 views

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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53 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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2answers
33 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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1answer
59 views

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
60 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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1answer
66 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
81 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
70 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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1answer
42 views

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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2answers
47 views

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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7answers
1k views

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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2answers
135 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 ...
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1answer
30 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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3answers
230 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
41 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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49 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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51 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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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 ...
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1answer
42 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 ...
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1answer
78 views

Differentiable at a point and invertible implies inverse is differentiable?

If $f:D\to\Bbb C$ is invertible and real (complex) differentiable at $c$ with $f'(c)\ne0$, it is easy to prove that if $f^{-1}$ is continuous at $f(c)$ and defined in a neighborhood of $f(c)$, then it ...
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36 views

Suppose A is $n$ x $n$ and the equation A $\vec{x} = \vec{b}$ has a solution for each $\vec{b}$ in $\mathbb{R}^n$

Explain why A must be invertible. Can someone explain why? I am a little confused here.
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60 views

Inverse Laplace Transform of $\frac{s}{(s-a)^{3/2}}$

Find the inverse laplace of: $\frac{s}{(s-a)^{3/2}}$ I tried working through this using partial fractions and convolution but I can't seem to get a requitible answer. How would I go about solving ...
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2answers
64 views

What is the inverse of $f(x) = a⋅e^{bx} + cx + d$

Does an inverse function for $f(x) = a⋅e^{bx} + cx + d$ exist where a, b, c, d are constants? If so, what is it? I've tried lots of methods, but they've all failed. What I ended up doing to ...
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1answer
67 views

Inverse transform of a modified Abel transform

I have been struggling for 6 months on finding the analytical inverse transform of a transformation below: $$F(y,k) = 2 \int_y^{\infty}\cos\left(ka\sqrt{r^2-y^2}\right) f(r,k) ...
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7answers
2k views

Why is the inverse of a sum of matrices not the sum of their inverses?

Suppose $A + B$ is invertible, then is it true that $(A + B)^{-1} = A^{-1} + B^{-1}$? I know the answer is no, but don't get why.
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72 views

Is every invertible matrix a change of basis matrix?

In the course that I am having, we are treating change of basis matrices as the matrices of the identity operation from one basis S to another basis say B. So, our instructor introduced a theorem : ...
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43 views

$\sigma-$algebra , inverse function

Is a $\sigma-$algebra a set that contains all the subsets of a set? In my lecture notes there isthe following: $$f(x)=\sin x \\ f^{-1}\left (\left [\frac{1}{2}, 1\right ]\right ...
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45 views

Cayley Transform and Eigenvalues

I have a particular operator, namely $A=-i\frac{d}{dx}$ that I would like to Cayley transform. $A$ is defined on the Hilbert space $L^{2}[0,1]$ and has domain $\mathcal{D}_{\alpha}=\{g:g \in ...
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0answers
31 views

{2}-inverse of a matrix - meaning of this notation

In "A Newton Method For Systems of $m$ Equations in $n$ Variables" by Yuri Levin and Adi Ben-Israel, authors use this notation $\{2\}$-inverse, however I am not familiar with it and thus I cannot ...
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2answers
116 views

Inverse Trig Functions

Find $f(x)$ if $f'(x)=4/\sqrt{1-x^2}$ and $f(1/2)=1$ So far I have integrated $f'(x)$ and have found: $$f(x) =y = 4\arcsin(x), x=4\sin(y)$$ $$1/2=4\sin(1)$$ $$1/2=4(\pi/2)$$ $$1/2=2\pi$$ So is ...
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2answers
62 views

Really confused about one-one,onto and invertibility.

I am really have some difficulty understanding how to do this problem. It asks to show that if T is one-to-one and onto, then T is invertible, and why T being invertible is equivalent to being one to ...
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1answer
30 views

does the vanishing of this determinant guarantee that 4 points lie on a circle or line?

while seeking to devise an intuitive, computation-lite demonstration of the fact that inversion preserves the concyclicity or collinearity of four points in the complex plane, i hit on the following ...
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Estimations for the size of the biggest entry in an inverse Matrix

If you got a Matrix $A$. Is there a estimation how big the largest element in the inverse of the matrix is? If it helps the matrix is unimodular, that means all entries are integer and the ...
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what is the inverse of this function

I'm weak at math and I need the inverse of this function if it's computable: $f(t) = A + (-2t^3 + 3t^2)(B-A)$ Note that $A$ and $B$ are constants. thanks for your help.
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6answers
70 views

How to find the multiplicative inverse of $2^{29} \mod 9$

I just started studying this topic and from my understanding I have to find an integer $x$ such that: $2^{29}x \equiv 1 \mod 9$ However, I have no idea of how to find a linear combination of $9$ and ...
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180 views

How to find inverse of 2 modulo 7 by inspection?

This is from Discrete Mathematics and its Applications By inspection, find an inverse of 2 modulo 7 To do this, I first used Euclid's algorithm to make sure that the greatest common divisor ...
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1answer
18 views

Inverse Fourier transform gives wrong results

If I have a polynomial: $a(x)= 2 + 5x -3x^2 +x^3$ The Fourier transform for N=4 is the evaluation of this polynomial in ${\omega}^0,{\omega}^1,{\omega}^2,{\omega}^3$ with ${\omega}^h = \cos(2\pi ...
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1answer
32 views

Taylor Expansion for an Inverse fonction

Is there a Straightforward Way to calculate taylor Expansion for the inverse function such as $$ \ e^{\sin x} $$ by knowing the taylor expansion for the function it self, I Think we use the ...
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1answer
51 views

When is the inverse of a sparse matrix dense?

The question is basically stated in the title. Say $A$ is a sparse square matrix, then Is there any way to estimate the density of non-zero elements of $A^{-1}$? What properties of $A$ are ...
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1answer
28 views

Matrices of the form $A=(x_i x_j)_{i,j}$

For some matrices that have a special form (e.g. Vandermonde matrices) there are simple, explicit expressions for their inverse available. A situation I ran into recently deals with matrices of the ...
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2answers
41 views

Gauss-Jordan elimation unstable?

Well for finding the inverse of any matrix (by hand) I learned to use the "Guass-Jordan elimination". However today I was looking it up again on wolfram|alpha. And what struck me is the line: ...
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2answers
64 views

Is there a way to simplify this inverse?

I have to compute an inverse of the form $$ (K + t(XO^T + OX^T))^{-1} $$ where $K,X,O$ are all $n\times n$, and $K$ is symmetric. Assume that $K$ is invertible and that $K^{-1}$ is known. I would ...
3
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1answer
67 views

Property of Matrix Inverse / Matrix Inverse Derivative

I am given real, symmetric matrices $X \succ B \succ A \succ 0$ (where '$\succ$' signifies positive definiteness such that if $B \succ A$ then $B-A \succ 0$ is positive definite). Further let the ...
2
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1answer
29 views

Inverse Matrices Where the Entries Are Variables

Is there a general formula, or a specific technique to find the inverse matrix of matrices where the entries are variables instead of numbers (is it even possible or defined)? For example, how does ...