Inverses include: multiplicative inverse of a number (reciprocal), inverse function, matrix inverse, etc. A subject tag such as (linear-algebra), (algebra-precalculus) or (arithmetic) should be added to clarify in which sense "inverse" is used. This tag should never be the only tag on a question.

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1answer
47 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 ...
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2answers
54 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 ...
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2answers
53 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
31 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 ...
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1answer
37 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 ...
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4answers
179 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 ...
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2answers
32 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 ...
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2answers
52 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
22 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
47 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
53 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
29 views

What is the inverse of the following function [closed]

what is the inverse of $$G(x):=\exp(-\exp(-x))?$$
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2answers
45 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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1answer
41 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
60 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
68 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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25 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
38 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 ...
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0answers
40 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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3answers
47 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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2answers
402 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
28 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?
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3answers
114 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
113 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
21 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
55 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 ...
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1answer
44 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 ...
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1answer
126 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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2answers
34 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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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 $f(x)$ ...
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1answer
57 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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1answer
32 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
40 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
44 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) = ...
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2answers
79 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
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2answers
39 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 ...
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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
26 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 ...
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1answer
108 views

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

I'm new to this site, Can someone explain if and if not
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1answer
53 views

If the function $f(x)=ax+b$ has its own inverse,then the ordered pair $(a,b)$ can be

If the function $f(x)=ax+b$ has its own inverse,then the ordered pair $(a,b)$ can be $(A)(1,0)\hspace{1cm}(B)(-1,0)\hspace{1cm}(C)(-1,1)\hspace{1cm}(D)(1,1)$ This is a more than one options correct ...
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12 views

Prove that $\frac{1}{[Z^{-1}]_{kk}}=\frac{\text{det}Z} {\text{det}Z_{kk}}=\text{det}Z_{kk}^{\text{SC}}$, $Z_{kk}^{\text{SC}}$ is the Schur complement

Suppose $Z$ is a complex (Wishart) matrix. Let $a=\frac{1}{[Z^{-1}]_{kk}}$, where $Z^{-1}$ is the inverse of $Z$ and $[Z^{-1}]_{kk}$ represents the $(k,k)$-th entry of $Z^{-1}$. When I was reading ...
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2answers
57 views

Finding the value of Inverse Trigonometric functions beyond their Real Domain

I wanted to ask how can we calculate the values of the inverse of trigonometric functions beyond their domain of definition, for example $\arcsin{2}$ beyond its domain of ...
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1answer
44 views

Can we always for an invertible matrix $M$ find real number $\alpha \neq 0$ such that $M+\alpha$ is invertible?

I do not know enough about matrices, maybe only enough to be able to create question like this one, but I would like to see an answer. Let $a_{ij}$ be some element of invertible $n\times n$ matrix ...
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1answer
28 views

Matrix Inversion distribution

How do you distribute the inversion in $(A^TA+\lambda I)^{-1}A^Ty$ assuming $A$ is a $n \times n$ square invertible matrix, $y$ is a vector with the dimension of $n$, and $\lambda$ is a constant?
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19 views

Implicit function theorem, what is the meaning of invertible linear operator?

I have to show that around $(1,-1,0)$(have to find the neighborhood as well), $x,y$ are determined uniquely by $z$ given $x+yz-z^3=1, x^3-xz+y^3=0$. What I did so far is: $f:\Bbb{R}^2\times\Bbb{R}\to ...
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1answer
20 views

Relation of numerical stability of matrix inversion and it's determinant

I have been taught that "inverting a square matrix with small determinant is numerically unstable because it is close to singular"? Is this right opinion?
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1answer
26 views

b is the inverse of a $( \mod 11)$

Let a and b be numbers in the set $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8 , 9, 10\}$ such that b is the inverse of a $(\mod11)$ and a and b are not equal. How many such subsets $ \{a, b\}$ of S are there?
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1answer
29 views

Production Model x=Cx+d — Use Inverse Matrix

Question: Consider the production model x = Cx + d for an economy with two sectors, where C= 0.0 0.5 0.6 0.2 and d= 50 30 Use an inverse matrix to determine the production level ...