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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55 views

Solving $\arcsin(\sqrt{1-x^2}) +\arccos(x) = \text{arccot} \left(\frac{\sqrt{1-x^2}}{x}\right) - \arcsin( x)$

If we have to find the solutions of equation $$\arcsin(\sqrt{1-x^2}) +\arccos(x) = \text{arccot} \left(\frac{\sqrt{1-x^2}}{x}\right) - \arcsin( x)$$ Using a triangle I rewrite it as $$2 \arctan ...
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41 views

Let A=$\tiny\begin{pmatrix}1&1&1\\1&2&2\\ 1 & 2 &3 \end{pmatrix}$ and B=$\tiny\begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 &0 \\ 1 & 1 &1 \end{pmatrix}$

Then (A) there exists a matrix C such that A = BC = CB (B) there is no matrix C such that A = BC (C) there exists a matrix C such that A = BC, but A $\neq$ CB (D) there is no matrix C such that A =...
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1answer
26 views

Laplace transform and inverse laplace transform

1- Find laplace transform for $4e^2t-3\cos^2(2t)+2\cosh(3t)$ My answer $L(4e^2t-3cos^2(2t)+2cosh(3t))=4L(e^2t)-3L(cos^2(2t))+2L(cosh(3t))$ $=\frac4 {s-2}-3L(\cos^2(2t))+\frac{2s}{s^2-9}$ But how ...
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1answer
24 views

symetric matrix inverse

Is there an easy way to invert a 3x3 symmetric matrix? for example A = $\begin{pmatrix} -1& 2& 0\\ 2& -5& 0\\ 0& 0& ...
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1answer
30 views

Verifying multiplicative inverses of modulo n are the elements that are relatively prime to n

A proposition in my book states: $(\mathbb{Z}/n\mathbb{Z})^{\times} = \{a \in \mathbb{Z}/n\mathbb{Z}~|(a,n) = 1\}$ which I want to prove. I start by defining $a$ in terms of prime factors $$a = p_1^{\...
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1answer
14 views

Find the inverse of $θ:P(\Bbb{Z})→P(\Bbb{Z})$ defined as $θ(X) = \bar X$

Find the inverse of $θ:P(\Bbb{Z})→P(\Bbb{Z})$ defined as $θ(X) = \bar X$ (the complement of $X$)? Would the inverse of the function just be the function itself?
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2answers
28 views

Changed codomain of inverse trigonometric functions

If codomain of $\arcsin(x)$ is $(\pi/2 , 3\pi/2)$ and codomain of $\arccos(x)$ is $(\pi , 2\pi)$ then what should be $\arcsin + \arccos$ equal to ? I thought of putting $x = \sin \theta$ But then ...
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1answer
47 views

Inverse of a “Vandermonde-like” matrix composed of power series

Is there an analytical formula for the inverse of a complex matrix whose elements are sets of "power series" except the last term is scaled? Let $0<x_1<x_2<...<x_n$ be monotonically ...
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1answer
50 views

Inverse function to $f(t)=3t+4ln(t+1)=y$

I have to invert the function $f(t)=3t+4\ln(t+1)=y$, so $f^{-1}(y)=t$. But I am struggling to invert this. Is there a solution?
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1answer
73 views

Prove $sgn(π) = sgn(π^{-1})$?

I'm pretty sure the inversion count of $π$ should be the opposite of the inversion count of $π^{-1}$. By this I mean if $π$ looks like this: $1 \to 1$, $2\to 2, \ldots, 10 \to 10$ and therefore the ...
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2answers
22 views

Simplifying Inverse Trig Function

I'm trying to figure out how to simplify this expression but I'm not quite sure on how to approach this question. How should I approach this question? Any help is greatly appreciated! $\tan(\sin^{-1}(...
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1answer
26 views

Show that a matrix satisfying certain conditions is non-singular

I have a square matrix $A$ satisfying the following conditions: The elements on the diagonal are negative; All other elements are non-negative; All row sums are less than or equal to $0$; There is ...
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1answer
27 views

I need someone to show me how to solve this input/output problem

Alright, so I have: $4y^3 = x$ And now I have to solve for $y$, where I can later use that equation to answer other questions I have. Can someone hint me out on how to solve for $y$ given the above ...
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0answers
18 views

The existance of Schur Complement Inverse

A block matrix $\mathbf{M}=\left[ \begin{array}{ccc} \mathbf{A} & \mathbf{B} \\ \mathbf{B}^T & \mathbf{C} \end{array} \right]$ is invertible if $\mathbf{A}$ and $(\mathbf{C}-\mathbf{B}^T\...
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3answers
39 views

Is the inverse of a real, continuous “1-1” function necessarily continuous itself? [closed]

If so, please do provide me with an epsilon-delta proof, if possible. Thanks in advance.
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1answer
25 views

Inverse of the sum of a invertible matrix with known Cholesky-decomposion and diagonal matrix

I want to ask a question about invertible matrix. Suppose there is a $n\times n$ symmetric and invertible matrix $M$, and we know its Cholesky decomposion as $M=LL'$. Then do we have an efficient way ...
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0answers
14 views

Harmonic inversion of an eccentric circle.

Inverted here is a circle with respect to another circle not as the conventional reciprocal inversion $ r_1 = \dfrac{a^2}{r_2}, $ but by means of a Lens formula known from time of Gauss: $$ 1/r_1 + 1/...
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1answer
21 views

Related to symmetric, diagonal and invertible matrices

While solving a problem I came across a specific question: Given $A,B$ as $2$ real, symmetric, matrices with $B$ positive definite, does there exist a matrix (invertible) $P$ such that both $P^TAP$ ...
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0answers
38 views

Finding the inverse of linear transformation using matrix

Assuming I have a linear transformation represented by a matrix with respect to some random bases, how could I find the inverse of the transformation using the matrix representation? I know I should ...
0
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2answers
28 views

Congruent matrices - why do we require invertiblility?

If $A$, $B$ $\in K^{n \times n}$ are $n \times n$ matrices over a field $K$, then we say that $A$ and $B$ are congruent if there exists an invertible $P \in GL(n, K)$ such that $B = P^TAP$, where $P^T$...
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0answers
57 views

Derivative of $(\lambda I - A)^{-1}$ with respect to $\lambda$

Is need to work with $\frac{d}{d\lambda} (1 - v^{T}(\lambda I - A)^{-1}u)$. Is it true that: $$\frac{d}{d\lambda} (1 - v^{T}(\lambda I - A)^{-1}u) = -v^{T}\frac{d}{d\lambda}(\lambda I - A)^{-1}u$$ ...
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0answers
15 views

How can I convince myself of the Fourier scaling property via inverse FT?

I have this function $f(at)$, and I want to Fourier-tranform it. I proceed in the following way, for $\quad\alpha<0 \Longrightarrow a=-|a|$: \begin{align} \ \mathcal{F}_{t \rightarrow \xi}[f(at)]= ...
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0answers
20 views

Find $v$ that maximises $v^TA\left(I_m-\sum_{i=1}^{p}B_i^Tvv^TB_i\right)^{-1}A^Tv$

I am trying to generate a rank 1 update rule for an optimization problem and I reach a point where I don't know how to continue. Let $v\in \mathbb{R}^n$ such that $\|v\|_2=1$ and also consider that $...
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1answer
46 views

Is there a way to update the inverse of a sum of two matrices following a rescaling of one of them?

Suppose I have two matrices $A$ and $B$ (let's assume that both $A$ and $B$ are invertible, as is their sum), and a scalar $g$. I am interested in the matrix $$M^{-1} = (A + gB)^{-1}$$ I am aware ...
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0answers
88 views

Inverse of the von Kármán-Trefftz transform

I'm having troubles finding the inverse of the Von Kármán-Trefftz transform (it's a conformal map) \begin{equation} z(x;b,k)=k\,b\,\frac{(x+b)^k+(x-b)^k}{(x+b)^k-(x-b)^k}\;,\quad b,k \in \mathbb{R},\;...
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1answer
27 views

Largest element in inverse of a positive definite symmetric matrix.

If I have an $n \times n$ positive definite symmetric matrix $A$, with eigenvalues $\lambda_{1}>\lambda_{2}\cdots>\lambda_{n}$, can I claim that the highest value which matrix $A^{-1}$ can have ...
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1answer
37 views

Each element is invertible [closed]

Let $R$ be a ring and let $I\subseteq R$ the only maximal right ideal of $R$. I want to show that each element $a\in R-I$ is invertible. $I$ is also an ideal. Could you give me some hints what I ...
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2answers
18 views

Finding Inverse in modulus m

I've been learning the Euclidean algorithm and came across this simple problem. $101^{-1} (mod 203)$ So I attempted it as such: $203 = 101(2) + 1$ So we got a gcd of 1, we can stop and do: $1 = ...
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1answer
27 views

Prove that an mxn matrix with m<n has no left inverse, similarly an mxn martix with m>n has no right inverse

I wonder how can I prove A matrix $A_(mxn)$ with $m \lt n$ has no left inverse and a matrix $A_(mxn)$ with $m \gt n$ has no right inverse Because I got no idea about that
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1answer
40 views

Inverse of cosh(x)

My goal is to find the inverse of $y=\cosh(x)$ Therefore: $$x=\cosh(y)=\frac{e^y+e^{-y}}{2}=\frac{e^{2y}+1}{2e^y}$$ If we define $k=e^y$ then: $$k^2-2xk+1=0$$ $$k=e^{y}=x\pm\sqrt{x^2-1}$$ $$y=\ln(x\...
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2answers
71 views

Question involving inequalities of greatest integer function of inverse trigo.

My question: Find the solution set of $$\lfloor sin^{-1}(x)\rfloor>\lfloor cos^{-1}(x)\rfloor$$ Can anyone help me to solve this question? Graphically it seems to be more complicated.
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3answers
41 views

How do you find the inverse of the function: $f(x)=-(5/3)x+10$

I'm given the following function $$f(x)=-(5/3)x+10$$ and told to find the inverse. By using mymathlab help and typing in the wrong number multiple times it shows me the answer is $$f^{-1}(x)=-(3/5)x+6$...
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2answers
24 views

Let $f = \frac{x}{x-1}$. What is ${f}^{-1}$? Show that f and ${f}^{-1}$ are symmetric about $y = x$.

Let $f = \frac{x}{x-1}$. What is ${f}^{-1}$? Show that f and ${f}^{-1}$ are symmetric about $y = x$. Finding the inverse of $f$ is easy enough. It actually turns out to be the same as $f$ itself. ...
0
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1answer
20 views

Inverses of Multivariable Functions from R^m to R^n

I was reading about Riemann Sphere and I found out that from the system: $$ u(x,y) = x/(x^2+y^2+1) \\ v(x,y) = y/(x^2+y^2+1) \\ w(x,y) = (x^2+y^2)/(x^2+y^2+1) \\$$ we can find inverses: $$ x(u,v,w)...
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0answers
23 views

Inverse fourier transform.

I am currently learning fourier transforms on my own, and I am stuck on a current conceptual idea that I need help with. I know that the $\int e^{-2\pi iyx}\,\mathrm{d}y=(2\pi)^{1/2}\delta(\omega-2\...
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1answer
34 views

Modulo multiplicative inverse of floating numbers

I have a floating value $k$ and an integer $P$ I want to calculate $(\dfrac{k}{\sqrt5}) \mod P$ How do I calculate it? PS: I know how to calculate MMI (Modulo Multiplicative Inverse of integer ...
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0answers
21 views

Inverted derivative question

I am wondering about the inverted function theorem $({\bf G}_{\bf F}({\bf x}))^{-1} = {1 \over {\bf G}_{\bf F}({\bf x})}$ If I have an equation ${\bf y} = \left({d {\bf F}({\bf x}) \over d {\bf x}}...
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1answer
42 views

Is a regular stochastic matrix definitely nonsingular?

Is a regular stochastic matrix definitely nonsingular (invertible)? How to prove it ? It says here that 'For a regular matrix always an inverse matrix exists' http://www.vias.org/tmdatanaleng/...
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2answers
31 views

Inverse Trig Functions Composite functions of Csc, Sec, And Cot

Ok guys.. I'm trying to get prepared for my test tomorrow and I did numerous exercises. But I stumbled upon one of the "types" of exercises. Which is a composite function in $\csc$ and $\sec$. For ...
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0answers
40 views

Matlab function for finding matrix inverse with cayley hamilton theory

I want to write function in matlab that would calculate the inverse of a matrix using its trace. I know that there are other ways to calculate the inverse but I need it to be with trace. I couldn't ...
0
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0answers
58 views

Find $\arctan(x) + \arctan(y)$ in terms of $\arctan((x + y)/(1 - xy))$ [duplicate]

Find $\arctan(x) + \arctan(y)$ in terms of $\arctan\left(\dfrac{x+y}{1-xy}\right)$. I want to essentially prove this equation given in the textbook: $$ \arctan(x) + \arctan(y) = \begin{cases} \...
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3answers
41 views

Homeomorphism & inverse, between $U=\{ (x,y) \in \mathbb{R^2} :|x|+|y|\leqslant 2 \}$ and $V=\{(x,y) \in \mathbb{R^2} : \max(|x|, |y|)\leqslant 3\}$

Find a homeomorphism, and its inverse, between $U$ and $V$ where: $U= \{ (x,y) \in \mathbb{R}^2 : |x|+|y| \leqslant 2 \}$ $V= \{(x,y) \in \mathbb{R}^2 : \max (|x|, |y|) \leqslant 3 \} $ I have ...
0
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1answer
54 views

Show that the special unitary group $SU(n)$ is a compact topological group

What I know: $SU(n)=${$A \in U(n): detA=1$} where $U(n)=${$n \times n$ matrices $A: AA^*=I=A^*A$} with elements in $\mathbb{C}$ and $A^*$ is the complex transpose of $A$ A topological group is a ...
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1answer
18 views

Analogue of right-inverse for non-surjective function

Given a function $f: X \to Y$, not necessarily surjective, is there a common name (and more concise definition than follows) for a function which maps elements in $Y$ where $f$ is defined to elements ...
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2answers
718 views

Example of a continuous function with a discontinuous inverse

What is an example of a function $f: \Bbb R^n \rightarrow \Bbb R^m$ such that $f$ is continuous and injective but that $f^{-1}$ is not continuous. Our professor teased us with the notion but I haven'...
1
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1answer
19 views

Having some trouble with inverse Laplace tranform

How to solve this using inverse Laplace transform? 1/[($s$+1)($s$+2)$^4$] I though of this solution which is $A$/($s$+1) + $B$/($s$+2) + $C$/($s$+2)$^2$ + $D$/($s$+2)$^3$ + $E$/($s$+2)$^4$ Then ...
0
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0answers
36 views

Inverse Laplace Transform of an Infinite Sum

How to find the Inverse Laplace Transform of the following expression $$1+\frac{-Xs^{2/a}-Ys^{3/b}}{1!}+\frac{(-Xs^{2/a}-Ys^{3/b})^2}{2!}+\cdots$$ Any approximation is also okay... Here $a$ and $b$ ...
0
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1answer
19 views

Differing graphs for simple inverse exponential problem

In class, we are learning exponential functions. The following inverse exponential problem is bothering me: $y=x^{-\frac{1}{9}}$. When graphed, I feel that it should look like it does on Desmos: ...
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0answers
29 views

Is my proof correct? (prequel to inverse matrices)

This question comes from a section before inverse matrices are introduced. Suppose $AD=I_m$. Show that for any b in $R^m$, the equation $A$x$=$b has a solution. [Hint: Think about the equation $AD$...
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2answers
16 views

Inverse calculation

I am trying to project estimated internal resistance of a battery. We know that the internal resistance approximately halves as the capacity of the battery doubles. For example... A 2AmpHour cell ...