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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52 views

Can the system $x+y=3$, $2x^2 + y^2 = 5$ be solved using matrices?

$$ x+y=3 $$ $$ 2x^2 + y^2 = 5 $$ I solved it by substituting $x = 3- y $ $2(3-y)^2 + y^2 =5 $ therefore, $ y= 2+\frac{i}{\sqrt3} $, $y= 2-\frac{i}{\sqrt3}$ However, I want to know that can I ...
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2answers
65 views

Suppose we have functions $f:A→B$ and $g:B→C$. Prove that if $f$ and $g$ are invertible, then so is $g \circ f$.

Suppose we have functions $f:A→B$ and $g:B→C$. Prove that if $f$ and $g$ are invertible, then so is $g \circ f$. Is the converse true? I.e., if $g \circ f$ is invertible, does it follow that $f$ and ...
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5answers
279 views

Is there a difference between $(x)^{\frac{1}{n}} $ and $\sqrt[n]{x}$?

Is there a difference between $(x)^{\frac{1}{n}}$ and $ \sqrt[n]{x}$ ? I'm confused with this topic. Any ideas or examples ? If $(x)^{\frac{1}{n}} = \sqrt[n]{x}$ Consider $x=\frac{-b \pm ...
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2answers
86 views

Inverting a matrix in $\mathbb{Z}/n\mathbb{Z}$.

So in my Linear Algebra course I was shown that we cannot directly use row reduction to invert a matrix over a commutative ring in general because the algorithm requires elements to be invertible ...
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2answers
34 views

Finding the inverse function of $f(x)=\frac{3x+1}{2-7x}$

Find the inverse function of: $$f(x)=\frac{3x+1}{2-7x}$$ I did the question and when I checked my answer with the key it was wrong, can someone please show me how to properly do this problem? I ...
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0answers
17 views

Inverse of a vector function

How i can prove that this map $f (\theta,\phi)=(\sin (\theta)\cos (\phi), \sin (\theta)\sin (\phi) ,\cos (\theta))$ with $0 <\theta <\pi , 0<\phi <2\pi$, has continuos inverse?
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1answer
44 views

Proof: Inverse of a Matrix

I know that to find the inverse of a matrix, I need to divide 1 by the determinant of the matrix followed by multiplying it by the adjugate of the matrix. However, what is the proof for this? I know ...
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1answer
23 views

Can this diagonal matrix be similar to it's negation?

Suppose I have a diagonal matrix such as $$ A:= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \\ \end{bmatrix}, $$ Is there a ...
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2answers
40 views

How to Integrate $1/\left(x\sqrt{1-4x^2}\right)$?

How do I integrate: $$\frac 1t\sqrt{1-4x^2}$$ I am thinking about (Integrals of Inverse Hyperbolic Function): $$\frac{-1}{a} \operatorname{arcsech}\left(\frac xa\right)+C$$ But do I need to use ...
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23 views

Need help proving that a matrix with a specific structure is non-singular

I am working on an engineering problem that requires finding the solution to the following system of linear equations: $$ \underbrace{\left(AB+C\right)}_M\boldsymbol{k}=\boldsymbol{y}$$ where ...
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0answers
44 views

Inverse of block triangular matrix

How to find the pseudo-inverse of the following block lower triangular matrix? $$X=\begin{bmatrix} A & 0 \\ c & d \\ \end{bmatrix}$$ Where $A$ is a $n\times n$ lower triangular matrix, $d$ is ...
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1answer
53 views

Confusion with orthogonal matrices

Here are two theorems from my textbook: https://imgur.com/a/UkDUD Why would the projection of y onto W not just be y times the identity matrix? Since an orthogonal matrix times its transpose is the ...
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0answers
118 views

Linear transformation is bijective if it has an inverse

$T : R^{n\times m}\to R^{m\times m}$, $C \to (AC)^T$ I need to find the inverse of this linear transformation, when A is an invertible matrix... Can anyone help me with this problem? I am asked to ...
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1answer
21 views

CDF RV Let Y = X^2. Compute the pdf fY (y). Since X = pY ,

Let $X$ be a continuous random variable with pdf $$f_X(x)=\begin{cases}6x^{-7}&\text{if }x\ge1\\0&\text{otherwise.}\end{cases}$$ Let $Y=X^2$. Compute the pdf $f_Y(y)$. Since $X=\sqrt ...
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0answers
22 views

Properties of frame operator of a matrix

We know many properties of gram matrix $\mathbf{X^TX}$, but what are the properties a frame operator of a matrix i.e., $\mathbf{XX^T}$ and what it tells us about $\mathbf{X}$ ? By properties i mean ...
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1answer
14 views

Expand matrix identity?

What are the intermediate steps to show the following? $$ (I+P)^{−1}(I+P−P) = I−(I+P)^{−1}P $$ I'm looking at the lecture slides here: http://www0.cs.ucl.ac.uk/staff/g.ridgway/mil/mil.pdf
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1answer
26 views

Find $g'(\frac{-1}{2})$ and $g''(\frac{-1}{2})$

Let $f(x)=\frac{x^3}{x^2+1}$, and $g(x)$ is the inverse function of $f(x)$. Then $f(-1)=\frac{-1}{2}$ and $g(\frac{-1}{2})=-1$. Find $g'(\frac{-1}{2})$ and $g''(\frac{-1}{2})$. I have found ...
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2answers
64 views

Finding right inverse matrix

Given a $3\times 4$ matrix $A$ such as $$ \begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ \end{pmatrix} ...
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0answers
42 views

Proving whether the following are groups or not.

In each case, I am asked to decide whether the indicated pair is a group or not. If so, prove it; if not, show which group axiom fails. (a) $(\dfrac{1}{2}\mathbb{Z}, +)$ where $\dfrac{1}{2} ...
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2answers
46 views

How to show that $\sin^{-1}(x)$ is an increasing function?

I would like to show that $\sin^{-1}$ is an increasing function. In other words, I want to show that $$x_{1} \leq x_{2} \rightarrow \sin^{-1}(x_{1}) \leq \sin^{-1}(x_{2}).$$ I fail to see how to ...
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1answer
29 views

How to Find the Inverse of the Function 13cos(12x)+1

If the domain is between [0,pi/12], how would I get this answer? So far, I have tried the following: 1) Switch 'x' with 'y', so, x=13cos(12y)+1. 2) Try to get 'y' by itself. Therefore, ...
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1answer
38 views

What is $\arccos(\cos(77\pi/4))$?

I did the following steps to get my answer of $-\pi/4$. Subtract as many $2\pi$ as possible from $77\pi/4$, which gives $77\pi/4-18\pi = 5\pi/4$. This leaves the angle in the 3rd quadrant. I believe ...
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1answer
55 views

How to Find the Domain of The Inverse of (4(e^x)-5)/(25(e^x)+12)

These are the steps I have taken so far: In order to find the inverse of the function, I did the following steps: ...
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1answer
549 views

Is there a way to calculate the definite integral of inverse of a 5th degree polynomial?

I want to calculate the definite integral of inverse of a 5th degree polynomial. The problem is that the inverse of the polynomial cannot be calculated (by using Matlab). However without calculating ...
3
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1answer
56 views

Inverse Laplace transform of $\frac{\exp(\frac{\lambda s}{1 - 2s})}{(1 - 2s)^{k/2}}$ (MGF of noncentral chi-squared distribution)

I am trying to use the countour integral to calculate the inverse Laplace transform of the function $$F(s) = \frac{\exp(\frac{\lambda s}{1 - 2s})}{(1 - 2s)^{k/2}} \hspace{1cm}\mathrm{for} \hspace{1cm} ...
4
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1answer
107 views

Inverse Laplace transform of one complicated function

I want to ask the inverse Laplace transform of the following function: $$F(s) = \frac{1}{s \cdot (1 + a \cdot s)^{m} \cdot (1 + b \cdot s)^{m-k}} \cdot \Bigl[\exp{(\frac{- c \cdot s}{ 1 + b \cdot s } ...
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1answer
18 views

Calculate the inverse of a multi-variable function

How would I calculate the inverse of $F(x,y)=\left( \textrm{arctan} \left(\frac{ay}{x}\right),\frac{x^2+a^2y^2}{2a}\right)$ $a$ is a constant.
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42 views

If $X$ is a $n$ by $k$ real matrix, and we know that $X^{T}X$ is invertible, is $X$ invertible as well?

If $X$ is a $n$ by $k$ real matrix, and we know that $X^{T}X$ is invertible, we know that $rank(X^{T}X) = n$ by $n$. Can we say that $X$ is invertible as well and hence has rank $n$? thanks!
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1answer
28 views

Effective way to calculate the inverse (A+kB)^-1 with k changing and A, B fixed

I have a Simulink modell where I need to calculate $(A+c_k B)^{-1}$ in every time step with $c_k$ changing each iteration. Does someone know any more effective way to do it, instead of calculating a ...
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1answer
96 views

Derivative of Inverse Function from AP

An AP question asks: The function f(x) = x^5 + 3x - 2 passes through the point (1,2). Let f^-1 denote the inverse of f. Then (f^-1)(2) equals? The inverse of this function should be y^5 + 3y - 2. ...
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2answers
86 views

Inverse Laplace Transfrom of $s^{-1}e^{-a\sqrt{s} + b/s}$

I am trying to find the inverse Laplace transform for following function and it seems almost impossible for me to find the answer. Can anyone help me please with final answer and also the way to get ...
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4answers
37 views

Find the interval for which $2\arctan x + \arcsin \frac{2x}{1+x^{2}}$ is an independent of x?

I used formula and simplified the expression to $\Rightarrow 4\tan^{-1} x$
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1answer
109 views

How to find the inverse of an upper triangular matrix

I want to find the inverse of an upper triangular matrix in an efficient way. I googled a lot, but all the articles discussed about a lower triangular matrix. Is it possible to edit the matlab code ...
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1answer
32 views

Calculation of inverse function

My question is physics related but the problem itself is just math. I have an expression for a refractive index depending on the wave length $\lambda$: $$ ...
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1answer
50 views

Inverse of $x\log(x)$ for $x>1$

Let $y(x)=x\log(x)$ for $x>1$. Is it possible to write down the inverse function explicitly? Has this inverse function been named? (For example, the Bessel functions are "named" but cannot be ...
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3answers
88 views

Inverse of a matrix!

Let $I_n$ be the $n$ by $n$ identity matrix and $b$ and $c$ be two vectors in ${\mathbb R}^n$ such that $b^Tc\ne 0$. Then one can easily see that the $n+1$ by $n+1$ matrix $X$ defined as following $$ ...
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2answers
42 views

Derivative of the inverse of a matrix

I've seen in a scientific paper this equation: $\frac{\delta K^{-1}}{\delta p} = -K^{-1}\frac{\delta K}{\delta p}K^{-1}$ where K is a $n\times n$ matrix which depends on $p$. In my calculations I ...
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2answers
62 views

If $y=ax^be^{-cx}$ then $x=g(y)$, find $g$

I have this function $$y=0.384394\cdot x^{0.341429}\cdot e^{-0.004749 x}$$ Based on this function I would like to know how I can I get $x=g(y)$.
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1answer
32 views

Inverting functions containing ceiling

I have a recurrence relation for a countably infinite sequence that contains the integers divisible by 5 but not by 7. The relation I came up with is: $5((n-1) + \lceil \frac{n}{6} \rceil)$ The ...
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0answers
41 views

Order of the sum of elements of the inverse of a matrix

For each $T$, let $A_T$ be a $T\times T$ matrix of real numbers. let $e_T$ be the $T\times 1$ vector of ones. Assume that the sum of all entries of the matrix $A_T$ divided by $T^2$ is limited as $T$ ...
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0answers
33 views

Differential Equation for Inverse of a Function given the ODE for the function

Assume we have and ODE for a function $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ in the form of \begin{equation} \mathbf{J}f \times F(x)=D(\lambda_i)\times f(x) \end{equation} where $\mathbf{J}$ denotes ...
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1answer
73 views

How to estimate the size of the neighborhoods in the Inverse Function Theorem

Given a function $f:U \subset V\to W$ such that $\textbf{D}f(x_0)\neq 0$ for some $x_0$. How to estimate the neighborhood for which it's invertible? Assuming the second derivative exists and is ...
5
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1answer
159 views

Which graphs do have invertible adjacency matrices?

I would like to know if there is any class of graphs for which the adjacency matrices are invertible. At this moment I am aware of only the class of graphs $n K_2$ which is the disjoint union of $n$ ...
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0answers
19 views

Inverse function of a function of Laplacian Matrix

I have a function $f:\mathbb R^n \rightarrow \mathbb R^n$ defined by $f(\mathbf{X})=L(\mathbf{X})\mathbf{X}$ where $L(\mathbf{X})$ is a (nonlinear) Laplacian matrix of an undirected but of-course ...
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2answers
79 views

Find a formula relating $\operatorname{arcsin}(x)$ and $\operatorname{arccos}⁡(x)$

From the formula $\sin\left(\frac{π}{2}−x\right)=\cos x$, find a formula relating $\operatorname{arcsin}(x)$ and $\operatorname{arccos}⁡(x)$. I have figured out that the domain of $x$ is ...
2
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1answer
68 views

Inverse of a function's integral

The function $g$ is strictly positive. Let the function $f$ be defined as $$f(x) = \int_0^x g(u) du$$ Is there a way to express $f^{-1}(x)$ in terms of $g$?
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1answer
108 views

Help with Inverse Function and Composition of Functions?

I'm currently doing work on discrete mathematics in my free time and am having some difficulties with understanding some questions pertaining to Relations and Functions. To be specific, I'm stuck on ...
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2answers
77 views

Find the inverse of the $n\times n$ matrix whose entries are given by $a_{ij} = \max (i,j)$

The actual question on the past papers is "Let $n\ge 1$ be an integer and consider the $n\times n$ matrix $A$ whose entries are given by $a_{ij} = \max(i,j)$ for all $1\le i,j\le n$. Show that $A$ is ...
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2answers
265 views

Show that f(x)=e^x from set of reals to set of reals is not invertible…

Yes, this is my question... How can you prove this? That $f(x)=e^x$ from the set of reals to the set of reals is not invertible, but if the codomain is restricted to the set of positive real numbers, ...
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5answers
114 views

Prove that if $f$ is increasing then so is $f^{-1}$

Prove that if $f$ is increasing then so is $f^{-1}$, when $f$ is a one-to-one function. I'm having trouble figuring out how to get started with this question. I'm assuming it has something to do with ...