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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100 views

checking whether functions satisfy Inverse Function Theorem.

I've my exam tomorrow and this question is expected to come but donot know how to solve... Here's the INVERSE FUNCTION THEOREM stated in my notes: It says: Let $E\subseteq \mathbb R^n$ be open ...
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3answers
51 views

show that every rational number has one and only one multiplicative inverse

I am stumped and have no idea on how I prove this. I don't know what else to say. I am beyond lost.
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2answers
58 views

The inverse of AR structure correlation matrix / Kac-Murdock-Szeg ̈o matrix

I want to find the inverse of the following matrix: $$ R_{k-1}=\begin{pmatrix} 1 &\rho &\rho^2 &\cdots &\rho^{k-2} \\ \rho &1 &\rho &\cdots ...
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1answer
57 views

Proof that if $A$ is similar to $B$, then $B$ is similar to $A$

$A$ is similar to $B$ if there is an invertible matrix $S$ such that $B = S^{-1}AS$. Prove that if $A$ is similar to $B$, then $B$ is similar to $A$. So if $A$ is similar to $B$ then $B = ...
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49 views

Showing there is no invertible function $f: \mathbb{R} \to \mathbb{R}$

I'm wondering whether there is an invertible function $f: \mathbb{R} \to \mathbb{R}$ such that $f(-1)=0$, $f(0)=1$ and $f(1)=-1$. I think it's not but I'm missing a real proof. The easiest would be ...
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0answers
13 views

Sherman Morrison Formula for hermitian updates

I have a problem in which, in principle I can apply twice Sherman-Morrison formula but it seems to me that for this case, there should be a simpler solution so my question is "May the process ...
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1answer
45 views

How to get tangent of inverse of curve??

Ok so my question is. Let $ f(x)=(1/7)x^3+21x-1.$ and let y=g(x) be the inverse function of f. Determine all points on the graph of the inverse function g so that the tangent line is perpendicular to ...
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1answer
24 views

What will $A^+A$ and $A^gA$ actually or exactly get if $A$ is not invertible?

I know if $A$ is invertible then $A^{-1}$ is the inverse of $A$, and $AA^{-1}=A^{-1}A=I$. I just learnt the concept of Generalized inverses and Moore–Penrose pseudoinverse. For a matrix $A$ that is ...
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1answer
18 views

conditions for Gauss_jordan elimination with no pivoting

Please note that here is Gauss_jordan elimination which help us get inverse of A. I am wondering, is there any condition that it could work without pivoting? I try to prove this under column ...
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2answers
17 views

Help with proving matrix transpose and inverses.

I am really struggling with these type of proofs. Could someone please give me hints on how to prove them, I do know the basic properties of transpose and inverse. If $ \mathbf{A} $ is invertible and ...
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1answer
23 views

Quadratic Equation with Matrix [Prove Invertible]

The problem is: "The $2\times 2$ matrix A satisfies $A^2-4A-7I=0,$ where I is the $2\times 2$ identity matrix. Prove that A is invertible." The hint given is: "We are trying to a matrix that is ...
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2answers
30 views

How does one compute the inverse of the function $f$ that satisfies $f(3x-2) = x-1$? [closed]

The problem is: Given $f: \mathbb{R} \to \mathbb{R}$ such that $f(3x-2) = x-1$, find $f^{-1}(x)$. It would be great if you could help me on this one
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0answers
36 views

The converse of the inverse function theorem

The inverse function theorem: A continuously differentiable function $F=(F₁,...,F_{r+1})$ defined from an open set $U⊂ℝ^{r+1}$ into $ℝ^{r+1}$ is invertible at a point $z=(s₁,s₂,...,s_{r},s_{r+1})∈U$ ...
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0answers
17 views

Inverse of a 2x2 principal submatrix whose inverse is known

Let $H$ be a $n\times n$ symmetric positive definite matrix. What is the (computationally) quickest way to obtain $H_{ij}$, the $2\times 2$ matrix whose inverse is the principal submatrix of the ...
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1answer
49 views

Circle Equation Surjectivity

Consider the circular function $g:\mathbb{R}^{2} \to \mathbb{R}^{+}$, $g(x,y)=x^{2}+y^{2}$. Show that it is surjective and continuous. Note This post has been amended in accordance with the ...
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0answers
16 views

matrix in normal coordinates

Writing the matrix $ \begin{pmatrix} -\frac{k}{\gamma} & \frac{k}{\gamma}&0&0&0&\cdots&0&0&0&0 \\ \frac{k}{\gamma} &-2\frac{k}{\gamma}& ...
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1answer
15 views

Reverse map for an equation .

I don't know this is actually reverse mapping or what but i have following equation. $$x = \tanh(a \cdot b ) + c $$ How do I solve for $a$? Does it has anything to do with inverse hyperbolic ...
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0answers
29 views

Definition of inverse binomial distribution

I am trying to succinctly define the inverse binomial distribution. Not the normal approximation, but the real thing, which will be discrete. So far I have this: $F^{-1}(\alpha;N,p) = k,\ \ s.t.\ \ ...
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3answers
42 views

Inverse of finite squared matrices.

I've usually used that given a square matrix $A$ with determinant $\det(A)\neq0$, then its inverse $A^{-1}$ is the matrix that meets: $$A^{-1}A=\mathbb{I}$$ and $$AA^{-1}=\mathbb{I}.$$ However, ...
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2answers
65 views

Prove $m=n$ of function $F:\mathbb{R}^n\to\mathbb{R}^m$ which has an inverse

Let $F:\mathbb{R}^n\to\mathbb{R}^m$ have an inverse function ${F^{-1}}:\mathbb{R}^m\to\mathbb{R}^n$ .If $F$ is differentiable at $a\in R^{n}$ and $F^{-1}$ is differentiable at $b=F(a)\in R^{m}$, ...
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2answers
62 views

Proving $(ab)^{-1}=a^{-1}b^{-1}$ where $F$ is a field and $a,b\in F$.

Proving $(ab)^{-1}=a^{-1}b^{-1}$ where $F$ is a field and $a,b\in F$. One thing to note is $a^{-1}\ne \large\frac{1}{a}$ (same goes for $b$) in this instance as there could be fields where this isn't ...
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6answers
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What's the inverse operation of exponents?

You know, like addition is the inverse operation of subtraction, vice versa, multiplication is the inverse of division, vice versa , square is the inverse of square root, vice versa. What's the ...
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1answer
57 views

How to find the inverse of f?

$ f : A \rightarrow B $ where $ A = B = \left \{4,5,6,7 \right \} $ $ f = \left \{ (4,6),(5,5),(6,7),(7,5) \right \} $ Find $ f^{-1} $ I know how to find the inverse of $ f $ if it were ...
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4answers
72 views

Why is $f(x)^{-1}$ used to denote the inverse of a function, and not its reciprocal?

Function notation says that any operations applied to a variable inside the parenthesis are applied to the variable before it enters the function, and anything applied to the function as a whole is ...
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1answer
19 views

Change of variable pdf inverse function

I've been given the following problem: $f(x,y) = e^{-(x+y)}$ on intervals $x \ge 0$ and $y \ge 0$, and $f(x,y) = 0$ otherwise. I'm also given that $Φ_1(x,y) = \frac{x}{y} = U$ and $Φ_2(x,y) = x + y = ...
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39 views

finding intervals on which f is a continuous inverse

I'm having trouble wrapping my head around this problem. I'm given a function f(x) - x + sinx and told to find all the intervals on which f has a continuous inverse. I honestly really have no idea ...
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2answers
47 views

Finding only first row in a matrix inverse

Let's say I have a somewhat large matrix $M$ and I need to find its inverse $M^{-1}$, but I only care about the first row in that inverse, what's the best algorithm to use to calculate just this row? ...
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2answers
29 views

Inverse of a function containing the ceiling function over the natural numbers

I am wondering if there exists an inverse function for $\lceil{e^{x}}\rceil$ over the natural numbers. I don't think it is a trivial task to derive an inverse function for a function containing a ...
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1answer
43 views

How to find the inverse function involving the exponential function?

Given: $f(x)= \dfrac{e^x}{1+9e^x}$ , what steps would I take to find its inverse? I tried following the steps on finding the inverse of a normal function but I keep getting one of the variables to ...
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0answers
48 views

What is the inverse kernel to this integral transform

What is the associated inverse kernel to the integral transform $T$ defined by \begin{align*} (Tf)(u) & = \int_{-\infty}^{0} \hat{f}(s)\exp((2i\pi+c)us)\ ds + \int_{0}^{+\infty} ...
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2answers
47 views

Question concerning Preimage

Let $f$ be the map from $\mathbb{R} \to \{a,b,c\}$ defined by \begin{equation} f(x)=\begin{cases} a &\text{if} \quad x>0 \\ b & \text{if} \quad x<0 \\ c &\text{if} \quad x=0 ...
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2answers
49 views

How to show that $AX=B$ has unique solution for invertible matrix $A$

If $A$ is an invertible $n \times n$ matrix, show that $AX=B$ has a unique solution for any $n \times k$ matrix $B$. I'm not sure where to start. What I have is that, if $A$ is invertible then ...
2
votes
1answer
58 views

Integration of a function containing inverse trigonometric functions

Q. $$\int \sin\left\{2\tan ^{-1}\left(\sqrt{\frac{3-x}{3+x}}\right)\right\}dx$$ $\implies$ $$\int \sin\left\{\sin ...
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1answer
37 views

Show that sum of elements of rows / columns of a matrix is equal to reciprocal of sum of elements of rows/colums of its inverse matrix

Suppose $A=(a_{ij})_{n\times n}$ be a non singular matrix. Suppose sum of elements of each row is $k\neq 0$, then the sum of elements of rows of $A^{-1}$ is $\frac{1}{k}$. Let ...
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1answer
21 views

Finding Inverse of exponential function

$f(x)=\frac {e^{(x)}} {(1+2e^{(x)})}$ I'm having trouble finding the inverse of this function algebraically.
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1answer
45 views

(n x n) Matrix multiplying itself with its inverse to form the (n x n) identity matrix

Is it ok to say Matrix A, with it's inverse, form the Identity Matrix? Thanks
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1answer
57 views

Prove that $g \circ f$ is a one-to-one function

Let $f$ and $g$ be one-to-one functions such that the domain of $f$ is $A$, the range of $f$ is $B$, the domain of of $g$ is $B$, and the range of $g$ is $C$. Prove that $g \circ f$ is a one-to-one ...
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3answers
305 views

Proof of Matrix Norm (Inverse Matrix)

Show for any induced matrix norm and nonsingular matrix A that $$ \left\|A^{-1}\right\| ≥ (\left\|A\right\|)^{-1} $$ where $$ \left\|A^{-1}\right\| = ...
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votes
1answer
34 views

Inverse matrices properties.

I know about the properties of matrix multiplication for multiplication such as $A(BC)=(AB)C$. However I'm curious if $(AC)B$ would also have the same value. I'm asked to represent $A$ in terms of $B$ ...
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votes
3answers
67 views

Find the inverse with respect to the binary operation $a ∗ b = a + b + a^2 b^2$

A binary operation on $\mathbb{R}$: $a * b = a + b + a^2 b^2$ The neutral element I found to be $0$. Then I need to find an invertible element having two distinct inverses. I don't know where to ...
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2answers
41 views

If there is a mapping of $B$ onto $A$, then $2^{|A|} \leq 2^{|B|}$

If there is a mapping of $B$ onto $A$, then $2^{|A|} \leq 2^{|B|}$. [Hint: Given $g$ mapping $B$ onto $A$, let $f(X)=g^{-1}(X)$ for all $X \subseteq A$] I follow the hint and obtain the function $f$. ...
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1answer
49 views

Following flash, a camera's battery begins to recharge the flash’s capacitor, which stores electric charge given by $Q(t) = Q_0(1 − e^{−t/a})$ [closed]

(The maximum charge capacity is $Q_0$ and $t$ is measured in seconds). (a) Find the inverse of this function and explain its meaning. (b) How long does it take to recharge the capacitor to 90% of ...
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1answer
176 views

Linear algebra proof regarding matrices

I'd like a hint rather than a full solution. The problem I am considering is the following: $X$ is an $n\times m$ matrix $Y$ is $m\times n$ Show that $(I - XY)^{-1}\cdot X = X\cdot(I - ...
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2answers
23 views

Slight help with inverse trigonometry question

I apologize for the lack of LaTeX, i will try to learn LaTeX and update this question as soon as possible. I am having some trouble with an inverse trigonometry question and was hoping that someone ...
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2answers
44 views

for $k\neq 0, -1, 1$, find the inverse of the matrix

for $k\neq 0, -1, 1$, find the inverse of the matrix $$\begin{pmatrix} k&0&0\\ 1&k&1\\ -1&1&k \end{pmatrix}$$ how am I supposed to solve this? all I can think of is plugging ...
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0answers
33 views

Study the associative and commutative properties and neutral and inverse elements of these groups

Group m*n = max(m,n) on Z and N So i showed its associative by m,n,p in Z and (m*n)*p = max(m,n)p =max(m,n,p) And m(n*p) = m*max(n,p) = max(m,n,p) Commutative m*n = max(m,n) and n*m = max(n,m). I ...
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1answer
41 views

Field Proofs with Multiplicative Inverses

I've been staring at these two for a while and I can't come up with anything concrete to start. Hints on how to begin would be greatly appreciated, full solutions are not necessary (and preferably ...
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1answer
51 views

expansion of matrix inverse

I would like to invert a square matrix $L$. One can write it as a sum of two matrices, one containing the diagonal terms ($D$) and the other the off-diagonal ones ($A$). $$L = D+A$$ I would like ...
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1answer
22 views

transpose and inverse multiplication

Given: $$A_{(n,n)} , B_{(n,n)}$$ A and B are invertible, is it possible that : $$(A^t B^t)^{-1} A^{-1} B^{-1} = I$$ I guess no, should this be true only if the AB=BA= orthogonal matrix ?
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1answer
53 views

Closed form of the inverse of a function

Does anyone know what the analytic form of the inverse of $f(x)=e^x+x$? Thanks in advance