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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Determinant of complex matrix with almost constant lines

Let $0\neq c\in\mathbb{C}$. Take the matrix $$A_C=\begin{pmatrix} n&c&\dots&c&c \\ c&n&c &\dots & c\\ c &c & n &c &\dots\\ \vdots ...
3
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4answers
103 views

explanation of $ \frac{dy}{dx} = \frac{1}{\frac{dx}{dy}} $?

I'm studying about derivative of inverse function. The teacher in the video (https://www.youtube.com/watch?v=3ReOtNCYuBw) (at 9:00 minute) said this if a differentiable function, f has an inverse, ...
2
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3answers
40 views

Bezout's Identity proof and the Extended Euclidean Algorithm

I am trying to learn the logic behind the Extended Euclidean Algorithm and I am having a really difficult time understanding all the online tutorials and videos out there. To make it clear, though, I ...
1
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0answers
30 views

Invertibility, inverse, and line weight of big circulant matrices

I am generating a random square sparse binary circulant matrix, defined by its first row. The length of the matrix is 9857 bits, and each line contains 71 ones, the rest are zeroes. I need to ensure ...
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5answers
56 views

What is the Inverse function of $y = 10^{-x}$? Steps are appreciated.

What is the inverse of $y = 10^{-x}$? These are my steps for the problem. Step 1 $y = 10^{-x}$. Step 2 $x = 10^{-y}$ by inverse substitution. Step 3 $10^y(x) = 1$. Step 4 $10^y = ...
1
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1answer
26 views

Proof that $A^{-1}=adj(A)/|A|$

I know that inverse of a matrix is given by $adj(A)/|A|$ but I cannot prove it.Nor did I find the proof in my books.Can you guide me?
0
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0answers
29 views

Inverse of sum of matrices

Let $A,B$ be invertible positive definite matrices of the same size. My goal is to efficiently compute $(xA + yB + zI)^{-1}$ for many triplets of positive real numbers $(x,y,z) \in \mathbb{R}^3$. ...
2
votes
2answers
32 views

Proof of positive semi-definite matrix

Consider a matrix $X$ to be \begin{equation} X=P-PA^\top\left(APA^\top + Q\right)^{-1}AP, \end{equation} where $P\in\Re^n$ is a positive definite matrix, $A\in\Re^n$ is a non-singular matrix, ...
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0answers
34 views

Existence Invers of polynomial with integer coefficients

I want to find is true my guess in below: Let $f(x)$ is a nonzero polynomial with integer coefficients $-1,0,1$ and $f$ has degree $n$. I say if numbers of $-1$ and $1$ in coefficients be different, ...
1
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1answer
39 views

Theoretical question about rank and invertibility of a block matrix,

Let A and B be real matrices, A is symmetric, and B has at least as many columns as rows. $$ C= \begin{bmatrix} A & B^t \\ B & 0 \\ \end{bmatrix} $$ a) Prove ...
2
votes
2answers
50 views

Find the inverse and determinant of A=(aI +T),

where is $a\ne 0$, $T$ has rank-one and zero trace. I just verified that a rank-one matrix has at most one non-zero eigenvalue. Now since T is of rank-one and has zero trace, that means all of its ...
0
votes
2answers
52 views

$f'(x) = \sqrt{1-f(x)^2}$, then $(f^{-1})'(x) =$

Math StackExchange, long time reader, first time writer. I have a question on inverse differentiation. The question is: Suppose $f'(x) = \sqrt{1-f(x)^2}$, then $(f^{-1})' (x) = ?$ I had a similar ...
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2answers
29 views

Inverse Function That Includes Fraction [closed]

Nice inverse function I am struggling on, totally forgotten how to move the fraction over: $$g(x)= \frac 1 {x-2}+5 \qquad (x>2)$$ Find the inverse function $g^{-1}$ specifying the rule domain and ...
0
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1answer
34 views

inverse trigonometry

Could anyone explain whether arc tan(sin x) can be simplified to an algebraic expression ? How can we draw graph of Arc tan(sin x). I have done an exhaustive google search only to find nothing. Also ...
3
votes
4answers
170 views

Theoretical question about the rank and existence of an inverse of a block matrix,

Let A and B be two $n \times n$ square matrices with complex coefficients, and consider the $2n \times 2n$ matrix $M$ given by $$ M = \begin{bmatrix} A & A \\ A & B ...
0
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2answers
29 views

Proof that if gcd(e, φ(N)) > 1, then a multiplicative inverse does not exist.

I am attempting a two-part problem on proofs and I am stuck on the second part. I think I have answered the first part correctly. (Note: these proofs are RSA-related, hence the variables) Here is the ...
0
votes
2answers
50 views

For matrix $A$ and any invertible matrix $C$, $CA$ has zero diag. Prove that $A=0$

Let $A$ be a matrix so that for all invertible matrices $C$, the diagonal entries of $CA$ are all $0$. Prove that $A=0$.
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1answer
21 views

Matrix multiplication identity proof

How can I prove that $(PQ + I_N)^{-1}P = P(QP + I_M)^{-1}$ knowing that we have two matrix $P_{N \times M}$ and $Q_{M \times N}$. Thank you very much for help.
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2answers
35 views

The inverse of bounded operator?

Is the inverse of a bounded operator always bounded , if yes how to prove it ?
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2answers
45 views

Is the inverse of a bijective monotone function also monotone?

If $f$ is bijective and monotonic function is $f^{-1}$ monotonic? Here is my attempt at solving the question but I'm unsure wether it's the right way to proceed or not. Mathematical translation of ...
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2answers
23 views

What is the inverse of the following function [closed]

what is the inverse of $$G(x):=\exp(-\exp(-x))?$$
2
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2answers
43 views

Intuitive explanation of left- and right-inverse

I am reading about right-inverse and left-inverse matrices. According to theory if a matrix $A_{m\times n}(\mathbb{R})$ is full row rank, then it has a left-inverse. That is, $AC=I_{m}$. Similarly, if ...
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votes
1answer
39 views

Find the inverse, domain and range of $f(x)=\frac{1}{\sqrt{-2x}}$

The inverse I am getting is $f^{-1}(x)= \frac{1}{2x^{2}}$. The domain and range of $f(x)$ is $x<0$ , $y>0$. The domain and range of $f^{-1}(x)$ is $x>0$ , $y>0$ though. What am I doing ...
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1answer
46 views

General solution for matrix inverse

I don't know if anyone know about this, but solving gpcm(generalized partial credit model) requires the inverse of the matrix of the form below. in Mathetmatica langauge, ...
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2answers
59 views

The inverse function

The function $g :\mathbb{Q} → \mathbb{Q}$ is defined by $g(r) = 4r + 1$ for each $r \in \mathbb{Q}$. (a) Determine $g(\mathbb{Z})$ and $g(E)$, where $E$ is the set of even integers. (b) Determine ...
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0answers
61 views

Inverse function of sum of coth and tanh terms

In a publication I found an equation of the form $c_p = B + C \left( \frac{D/T}{\sinh(D/T)} \right)^2 + E \left( \frac{F/T}{\cosh(F/T)} \right)^2$ $c_p$ is the heat capacity, $T$ is the ...
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0answers
21 views

Inverse of character table

The character table of a group is always invertible, because the rows are orthogonal. Is there a general formula to compute the inverse of the character table?
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2answers
35 views

Algebra 2 - Find Domain and Range of Function and Its Inverse

$f(x)=-x^2+1$ For some reason, the inverse $f^{-1}$ gives me a domain equal to 1 or less than with a range of all real #'s. But the domain of the original function f(x) can only be negative. As ...
2
votes
0answers
29 views

Matrix inverse series expansion

I want to prove that when $I+K$ is invertible, $$(I+K)^{-1}=I-K+o(K)$$ to establish that the matrix inverse function has derivative $-I$ at $I$. My hope is that this identity carries over from ...
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vote
3answers
43 views

Given a diagram of $f(x)$, how do you find $f^{-1}(-1)$?

A question has a diagram of function f, with no function given (please ignore the purple line): If I wanted to find $f^{-1}(-1)$, would I draw the inverse of f along the x=y line and then find what ...
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votes
2answers
394 views

Inverse of block marix

Let $A, B, C, D$ be invertible square matrices. How can I find the inverse of $\begin{bmatrix} A & B \\ C & D \\ \end{bmatrix}$? Thank you.
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1answer
27 views

What does it mean for det($A^2-\lambda I_n)=0$ to have a zero solution, assuming $A$ is an invertible matrix?

If it means that $\lambda$ must be zero, then I think the statement is incorrect as det($A)\ne 0$. Could someone clarify the meaning?
3
votes
3answers
112 views

Finding an “inverse function” symmetrical to y=2x not y=x

Hello I'm very inexperienced in math (I know a little about derivation/integrals etc but nothing on university level) so my terminology will not be on point (as well due to english not being my native ...
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6answers
58 views

Multiplicative inverse of 47 mod 64.

I have to compute the multiplicative inverse of $47$ $mod 64$. What is the fastest way to do this?
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0answers
20 views

Inverse Fourier Transform of (e^w)*F(w)

If the inverse Fourier Transform of F(w) is f(t), what is the inverse Fourier Transform of (e^w)*F(w) ? My best guess is that I should expand e^w into a power series and use the fact that the inverse ...
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0answers
30 views

Inverse function of hypergeometric function, e.g., ${}_{2}F_{1}(1,1;1.2;x)$

I want to know whether it is able to express the inverse function of hypergeometric function using some special function. For instance, the Gauss hypergeometric function ...
0
votes
1answer
35 views

Prove or disprove the following problem. [closed]

If $A$,$B$ and $A+B$ are non-singular matrices then $A^{-1}+B^{-1}$ is non-singular and $(A^{-1}+B^{-1})^{-1} = A(A+B)^{-1}B$
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3 views

Leveraging the inverse in nonlinear optimization

Consider a non-linear optimisation problem like $$\mathcal{L} = \left\|{\bf x} - f({\bf y})\right\|$$ which we aim to minimise for vector ${\bf y}$ and where $f(.) : \mathbb{R}^N \mapsto ...
4
votes
1answer
121 views

Inverse of $f(x)=3^x+2^x$

I'm tring to find inverse of $f(x)=3^x+2^x$ but I don't have any clue. I tried to $$y=2^x((3/2)^x+1)$$ $$\ln y=\ln2^x+\ln((3/2)^x+1)$$ $$\ln y= x \ln2+\ln((3/2)^x+1)$$ but I can't continue
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votes
2answers
32 views

Given $2\arcsin(x)-3\arccos(x)=\frac{\pi}6 $, find the value of x.

I know that $\arcsin(x) + \arccos(x) = \frac{\pi}2$, but how to use that to solve the following question? $$2\arcsin(x)-3\arccos(x)=\frac{\pi}6 $$
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votes
1answer
19 views

Using inverses to find solutions.

So upon solving some trigonometric equations, I found myself using the following method often:$$f[g(x)]=h(x)$$$$f[g(g^{-1}(x))]=h[g^{-1}(x)]$$$$f(x)=h[g^{-1}(x)]$$Which is how I usually find $f(x)$ ...
2
votes
1answer
54 views

Finding the inverse of a “bow-shaped” matrix

Consider the matrix $$A = \begin{bmatrix} n_{+} & n_1 & n_2 & n_3 & \cdots & n_{r-1} \\ n_1 & n_1 & 0 & 0 & \cdots & 0 \\ n_2 & 0 & n_2 ...
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vote
1answer
30 views

Jacobian of inverse of matrix $A(x) \in \mathcal{M}_{7\times7}$?

I have a matrix $A(x)$ where $x\in \mathbf{R}^{7}$. I have to calculate $\frac{\partial}{\partial x}A(x)^{-1}$ and then I will evaluate it at some $x_{0}$. Now this matrix is very dense so its not ...
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0answers
33 views

Derivative of inverse matrix

Suppose $\Omega \left( \mathbf{\alpha }\right) $ is a $T\times T$ full rank matrix where $\mathbf{\alpha }$ is a $p\times 1$ vector, then what's the exact expression for $\frac{\partial \Omega ...
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0answers
40 views

Inverse Laplace transform of complicated function

I have a Laplace transformed function that I'd like to transform back. It's quite a complex function however, which is why I am stuck: $$C(x,s) = ...
9
votes
2answers
78 views

Prove that $\ln$ and $\exp$ are inverses

If we take the definitions of $\exp$ and $\ln$ as follows: $\exp(x) = {\large\sum\limits_{i=0}^\infty} \dfrac{x^i}{i!}$ $\ln(x) = {\large\int_1^x} \dfrac1t\ dt$ how could we prove that these ...
2
votes
2answers
34 views

Proof about Diagonalization of A

The question asks WHY is it true that $$A^{n} = PD^{n}P^{-1}$$ I can never do proper proving in algebra; what I almost know for sure is that a proof by induction is the way to go here. But how do you ...
0
votes
1answer
56 views

$2 \ne 3$, but where's my error?

In $\mathbb{Z}_6$, $3^3 = 3^{-3}$ since $3^{-3} = 3^{6-3} = 3^3$. Thus $(3)^3 = (3^{-1})^3=2^3=2$. But also $3^3 = 3$ in $\mathbb{Z}_6$. Where's my error? Sorry for this question, but I think I got ...
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1answer
22 views

Inverse function on given sets

My question is: Given sets $A$ = {$a_1, a_2$} and B = {$b_1$}, let the function $f$ from $A$ to $B$ be given by the following set of ordered pairs, $f$ = { ($a_1, b_1$), ($a_2, b_1$) }. If $f$ has an ...
2
votes
1answer
81 views

Checking if two matrices are inverses of each other. [closed]

I'm new to this site, Can someone explain if and if not