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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Finding Inverse of a matrix using elementary transformations

So I have to find the Inverse of A. $$ A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 3 & 4 \\ 3 & 4 & 3 \\ \end{bmatrix} $$ By using elementary row or column transformations.. The ...
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41 views

Numerical Algorithm for $n \times n$ Matrix Inverse

I have to write a C program in which I have to compute the matrix inverse of a $n \times n$ matrix. Is there a convenient iterative process that I can use to do that? All I see is the co factor method ...
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1answer
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Special Case Linear Solvers

I, and friends of mine, are interested in matrices which can be inverted / solved easily (i.e. in less than O(n^3)). I started to put together a github page dedicated to it and so far have identified: ...
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Invariance of Frobenious norm under transformation.

Can we say for every invertible square matrix $\mathbf{P}$, $\Vert\mathbf{X-B}\Vert_F^2=\Vert\mathbf{P^{-1}(X-B)}\Vert_F^2$. And does this hold true for non-square matrix $\mathbf{P}$ under some ...
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4answers
74 views

How does $1 + \tan^2x = 1/\cos^2x$?

I am unable to see why $$1 + \tan^2 x= 1/\cos^2x$$ I have looked into the topic anad I am familiar with the reciprocal ratios of cosec, sec, and cot. but cannot derive how this statement makes sense. ...
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1answer
20 views

Having a holomorphic function $h$ that's the inverse of a function $f$, it's also the inverse for a continuation of $f$

Let $\gamma: [0, 1] \to \mathbb{C}$ be a (continuous) path, $\gamma(0) \in D$, $(f, D)$ a tuple of a holomorphic function $f: D \to \mathbb{C}, D \subseteq \mathbb{C}$ a simply connected open set. Let ...
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2answers
46 views

Proving matrix properties: [closed]

Prove: (i) $A(I+BA)^{-1}=(I+AB)^{-1}A$ (ii) $(I+AB)^{-1}=I-A(I+BA)^{-1}B$ (i) Consider $A(I+BA)=(A+ABA)=(I+AB)A$ Taking inverse on both sides (invert) ...
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1answer
47 views

Using Chinese Remainder Theorem to find an integer $x$ for which $ x\equiv 3\pmod 4 x\equiv 5\pmod 9 x\equiv 10\pmod {35} $

Hello I have got problems with understanding the reduction method in CRT. We have got system like this $$x\equiv 3\pmod 4$$ $$x\equiv 5\pmod 9$$ $$x\equiv 10\pmod {35}$$ There is a way to do this ...
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1answer
57 views

Inverse of matrix with particular structure

I have a square invertible matrix $A=[c, c^2, c^3 \dots c^n]$ where $c \in \Bbb R^n$. Are there any known fast tricks for inverting it? Edit: $c$ is a column vector and raising it to a power is to ...
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46 views

What approximations for the Gamma function's inverse appear to work 'best'?

So I was wondering how we approximate the inverse of the Gamma function, where I tried a few methods: Lagrange inversion theorem: $$\Gamma^{-1}(z)=a+\sum_{n=1}^{\infty}\lim_{w\to ...
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1answer
28 views

Inverse trigonometric expansion related question

I know expansions for $\sin^{-1}(x)+\sin^{-1}(y)$, but does there exists any expansion for $\sin^{-1}(x \pm y)$ if not then what is the reason?
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2answers
70 views

How do I prove that $(ABC)^{-1} = C^{-1} B^{-1} A^{-1}$ [closed]

Please help me answering this problem! thank you :) Prove that for any nonsingular matrices $A$, $B$, and $C$, the equation $$(ABC)^{-1} = C^{-1}B^{-1}A^{-1}$$ holds. (Hint: Assume $D$ is the ...
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2answers
31 views

what condition of A makes transpose(A)*A nonsingular?

What contidion of A makes $$A^TA$$ nonsingular? If so, that is $$A^TA$$ is non-singular than a unique solution exists.
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25 views

partially ordered group, does x=-x imply x=0?

I have just a simple question: Let (G,+) be a partially ordered Abelian group. Does x = -x imply x = 0 ? If the answer is yes, then how could i prove it? If the answer is no, then a ...
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1answer
31 views

Find $\sinh^{-1}x$

The hyperbolic sine function, $\sinh(x)$ , is defined by the equation: $$ \sinh(x) = \frac {e^x-e^{-x}} {2}$$ Find a formula for its inverse, $$ \sinh^{-1}(x) $$
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1answer
30 views

Why arsin function has range $[-\pi/2,\pi/2]$ [duplicate]

While studying in P.75 of inverse trigonometric functions it tells we have to restrict our domain before finding the inverse.But I can't get why we choose $[-\pi/2,\pi/2]$?Why can't we choose ...
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1answer
13 views

Ensure that for each number in specific space there is inverse

Let say I want to find the Inverse number of some serial number. ( 9 digits number .. its can be an ID). And let say we want to find the inverse in $\mathbb Z_{1000000123}$ ( for example ) How I can ...
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1answer
115 views

How to convert $(A+\lambda E)^{-1}$?

Here is one of the most famous equation called Sherman–Morrison formula (1951) when we want to get an inverse matrix. ...
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1answer
69 views

Getting $B$ from $A = M^t B M$ without inverting $M$

I have got three matrices: $A$ (dimension $n \times n$), $B$ (dimension $m \times m$) and $M$ (dimension $m \times n$). We have $m > n$. This is the relation between these three matrices: $A = M^t ...
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153 views

How to invert this expression involving $\tanh^{-1}$?

I've got the expression: $ x = \tanh^{-1}(p) - \sqrt{\frac{2}{3}} \tanh^{-1}\left( \sqrt{\frac{2}{3}} p\right) $ How can I invert this function so I have a function $p(x)$? I thought about using ...
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2answers
95 views

How to find inverse of below function $y=2^x+3^x ,\ y^{-1}=?$ [closed]

I need to find inverse of below function: $$y=2^x+3^x ?$$
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21 views

Why is the range of inverse cotangent only positive? [duplicate]

Is it because of conventions or some practical reason? Some websites say it is conventions while other websites say it is because of making it a function. Thanks!
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36 views

Computing one-sided inverse of a matrix over some finite field

Let $M$ be a $k\times n$ matrix with $k < n$, and assume that $\text{rank}(M)=k$. Over $\mathbb{R}$, one can compute a right inverse of $M$ as follows: $$M_\text{right}^{-1} = M^T(MM^T)^{-1}$$ ...
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35 views

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 ...
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4answers
107 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, ...
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3answers
61 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 ...
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44 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
60 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 = ...
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1answer
29 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?
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38 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 $(x,y,z) \in \mathbb{R}^3$. ...
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2answers
43 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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1answer
48 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
57 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
38 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
181 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
33 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
48 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
55 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
47 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
61 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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69 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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27 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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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 ...