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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Find inverse of a double function

I have the following function: $$f(x)=\begin{cases}3x+1,~x\gt 0\\2-x^2,~x\leq 0\end{cases}$$ and I need to find its right inverse. So far I got that, ...
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28 views

Prove that $\det(A) \cdot v \, A^{-1} = \det(A+uv) \cdot v \, (A+uv)^{-1}$.

Let $A$ be a $n \times n$ matrix, $u$ a $n \times 1$ matrix and $v$ a $1 \times n$ matrix. If $A$ and $(A+uv)$ are invertible, prove that $$ \det(A) \cdot v \, A^{-1} = \det(A+uv) \cdot v \, ...
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1answer
671 views

Proof that Gauss-Jordan elimination works

Gauss-Jordan elimination is a technique that can be used to calculate the inverse of matrices (if they are invertible). It can also be used to solve simultaneous linear equations. However, after a ...
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2answers
24 views

Why does the Gaussian-Jordan elimination works when finding the inverse matrix?

In order to find the inverse matrix $A^{-1}$, one can apply Gaussian-Jordan elimination to the augmented matrix $$(A \mid I)$$ to obtain $$(I \mid C),$$ where $C$ is indeed $A^{-1}$. However, I fail ...
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63 views

The inverse Laplace transform of $ s^{3/2}-a-bs \over s^{3/2}+a+bs$

How can I solve the inverse Laplace transform as below: $$\mathscr{L}^{-1}\left( s^{3/2}-a-bs \over s^{3/2}+a+bs \right) $$ where a and b are constants. Hint: we can consider $${ s^{3/2}-a-bs ...
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1answer
19 views

Searching for a function where the inverse exists in a neighborhood of a point, but the Jacobian is zero.

I'm looking for a function f in $\mathbb{R}^2$ such that the inverse function therem at some point P = (x,y) does not give an answer of whether the function is invertable in some neighborhood of P, ...
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1answer
65 views

Find the multiplicative inverse of $5$ in $\mathbb Z_{73}$

I'm having some trouble with this question. The inverse should result in $44$ but I am getting $29$ $$73 = 14 \times 5 + 3$$ $$5 = 1 \times 3 + 2$$ $$3 = 1 \times 2 + 1$$ so $\gcd(73,5)=1$ using ...
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16 views

Problems with the inverse of a banded matrix: not invertible?

I am creating with a software a banded matrix, which is also symmetric. In fact, its definition comes from an array, Array[q], whose length is ...
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21 views

Invert an Excel function containing the tangent

In the following excel formula: =95*1*1/TAN(RADIANS(M3-(10.3/2.01)))/5280) $M3=2.63715$ and let's say the result of this formula is: $5.508306483$ What would ...
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958 views

convexity of inverse function

I have a question on the reverse of a convex function. Let $f(x)$ be a convex function. Is the reverse function, say $g(x)=f(x)^{-1}$, is necessarily a concave function ? Considering that such ...
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34 views

Use Euclid's algorithm to find the multiplicative inverse $11$ modulo $59$

I was wondering if this answer would be correct the multiplicative of $11$ modulo $59$ would be $5$ hence $5\cdot11 \equiv 4 \pmod{59}$. Is this correct?
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44 views

Integration.Matrix.Determinant.Inverse.Trace.

Given $$ I_n=\int_0^1\frac{x^n}{x^{2012}-1}{\rm d}x\text{ and }J_n=\int_0^1\frac{x^n}{x^{2013}+1}{\rm d}x\quad\forall n>2012, n\in\mathbb N$$ If the matrix $$\rm A=[a_{ij}]_{3\times3}\text{ where ...
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1answer
28 views

Showing multiplicative inverse has the same order $\pmod{p}$?

Suppose that $a$ has order $h \pmod{p}$ and $a\overline{a} \equiv 1 \pmod{p}$. Show $\overline{a}$ also has order $h$. I'm a little confused as to how to start proving this -- I know that if $a$ has ...
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3answers
36 views

What does inversion mean?

I am in highschool taking some advanced math courses and I have some questions about terminology. There appears to be more definitions to the meaning of inversion in math than I can count. I'm ...
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1answer
46 views

Questions on the formula for 2x2 inverse

Hi I was working on inverting 2by2 matrix in general form by using a,b,c,d. I know the formula (which is below) but I have questions in the process of getting the formula. 1) To get rid of the ...
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33 views

Inverse Matrix in 3D

Suppose we have the following matrix in three dimensions $$ M_{ij} = g_{ij} + e_{ijk}z^{k} $$ where $e_{ijk}$ is an antisymmetric density, i.e. $e_{ijk} = \sqrt{\det g}\cdot\epsilon_{ijk}$ and $z^{k}$ ...
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3k views

Finding the inverse of a matrix by elementary transformations.

While using the elementary transformation method to find the inverse of a matrix, our goal is to convert the given matrix into an identity matrix. We can use three transformations:- 1) Multiplying ...
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48 views

If two invertible matrices A and B commute, then A^-1 and B^-1 must commute as well ??

If two invertible matrices A and B commute, so their inverse must commute as well or not ?
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19 views

Oracle for the inverse function

Let $F$ be a 1-1 function from $[0,1]$ onto $[0,1]$, which is continuous and monotonically increasing. Two oracles are given: A direct oracle - given $x\in[0,1]$, it returns $F(x)$. An inverse ...
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34 views

How to calculate inverse of Variance Gamma call price formula using Newton-Raphson search

The Variance Gamma call price formula is given by: $$C(0)= \int\gamma(R) e^{-rT} \int f\left(S(0) e^{\theta R+\omega T+\frac12 \sigma^2 R} e^{rT-\frac12 \sigma^2 R+\sqrt{T}\sqrt{R/T} \sigma ...
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1answer
23 views

Derivative of inverse cosecant?

I am slightly confused by this, because when I worked out the derivative of arccosec(x), my answer was $\frac{-1}{x\sqrt{x^2-1}}$, which agrees with the answers online. However this would imply that ...
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1answer
284 views

Inverse of the sum of the inverse of two matrices

I need to compute $ (A^{-1} + B^{-1})^{-1} $. Both $A$ and $B$ are symmetric and $A$ is invertible and PSD. I already know $B^{-1}$ and $A$, but I don't have $A^{-1}$ and $B$. Is there a formula to ...
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1answer
50 views

Relation between two inverses

Suppose you know $(I+T)^{-1}$, is there any way for approximate the inverse of the matrix $(I+\alpha T)^{-1}$, where $\alpha\in{\mathbb{R}}$?
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38 views

how to plot the graphs of $\sin^{-1}(\sin x)$,$\cos^{-1}(\cos x)$,$\tan^{-1}(\tan x)$

Can someone please explain me how to plot the graphs of $\sin^{-1}(\sin x)$,$\cos^{-1}(\cos x)$,$\tan^{-1}(\tan x)$ ? I am having a little difficulty in understanding how the nature of the graphs can ...
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2answers
425 views

Inverse functions and tangent line

Let $f(x) = \frac14x^3 + 12x + 6$ and let $y = f^{-1}(x)$ be the inverse function of $f$. Determine the $x$-coordinates of the two points on the graph of the inverse function where the tangent line is ...
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2answers
13 views

Inverse of matrix with 1 in diagonal and some entries above them.

Suppose matrix N has a,b,c above the main diagonal, and all other entries equal to $0$. that is, $N=\begin{bmatrix} 0 & a & 0 & 0 \\ 0 & 0 & b & 0 \\ 0 & 0 & 0 & c ...
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22 views

Inverse function for a surface of revolution

I have the following function: $$ f(x)= c_1\cdot c_2\cdot x\cdot \arctan\left(c_2\cdot x\right)-\frac{1}{2}\cdot c_1\cdot \ln\left(1+c_2^2\cdot x^2\right) $$ with $c_1=0.003$ and $c_2=150$ constants ...
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68 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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32 views

Evaluate cos[(1/2)[arcsin(-3/5)]]. I'm not sure what i'm doing wrong.

$x=\arcsin(-3/5), \; \sin x = -3/5$ **Drew a triangle to find $\cos x$ $\cos x = 4/5$ Now, I don't know what to do from here. I know I have to use a double angle formula, but when I evaluate the ...
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924 views

A problem with the geometric series and matrices?

Let $n$ be a positive integer. Let $A$ be a square matrix. Let $I$ be the identity matrix with the same size as $A$. I want to simplify $f_n(A) = I + A + A^2 + A^3 + A^4 + \cdots + A^n$ Now I know ...
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1answer
375 views

relation between size of matrix and condition number

I have a matrix A of size NxM. Is there any relationship between size of a matrix A with the condition number ? I am computing the pseudo inverse (pinv in matlab ) ...
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47 views

a special matrix inverse

Let $A=\left( \begin{matrix} {{A}_{11}} & \ldots & {{A}_{1n}} \\ \vdots & \ddots & \vdots \\ {{A}_{n1}} & \cdots & {{A}_{nn}} \\ \end{matrix} \right)$ be an ...
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1answer
22 views

Inverse of function with two Exponential Eulers Terms

How can I go about getting the inverse of$ f(t) = e^{-.001t}\cdot e^{-.005t}$? I have found a couple of calculators online that suggest that the answer is: $t=-166.667\ln(y)$, but I would like to know ...
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2answers
39 views

Solving $z=w/2-\sin(tw)/(2t)$ for $w$

Is it possible to solve $$z=\frac{w}{2}-\frac{\sin(tw)}{2t},$$ for $w$? My first thoughts were that we would have to be careful about the domain of $f(w)$ so that the inverse was actually a function ...
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26 views

How to solve an inverse relationship (cooking temp/time)

How to figure out exactly the "add a little more time" to the question: cook at 425 deg for 18 minutes ... if I have several things in the same oven and need to set the oven at 375. I can't use a ...
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1answer
37 views

Matrix derivatives of determinant and inverse related to $\mathbf{X}\mathbf{X}^{T}+\mathbf{C}$

I would like to calculate the derivatives of determinant and inverse related to the term $\mathbf{X}\mathbf{X}^{T}+\mathbf{C}$ with respect to $\mathbf{X}$, where $\mathbf{C}$ is a constant matrix. ...
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1answer
33 views

help with inverse function in $\mathbb R^2$

$F(x,y)=(x^2+2y^2,2x^2+y^2)$, and $A=\{(x,y):x>0,y>0\}$ I need to show $F(A)=\{(u,v):0<u/2<v<2u\}$ I also need to find what is $G(=F^{-1}):B\rightarrow A$ For the first question I ...
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18 views

$F(x,y) = (x^2 + 2y^2, 2x^2 +y^2)$ and $A= \{(x,y) : x>0, y>0\}$

$F(x,y) = (x^2 + 2y^2, 2x^2 +y^2)$ and $A= \{(x,y) : x>0, y>0\}$ I need to find the following: $(a)$ Show $F$ is one-to-one on $A$. $(b)$ Show that $F(A) = \{(u,v) : 0 < \frac{u}{2} < v ...
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1answer
26 views

Inverse symmetric circulant matrix

I want to inverse a very particular matrix numerically. The matrix is always symmetric and circulant. As an example of a 4x4 matrix I would want to inverse \begin{pmatrix} v_0 & v_1 & v_2 ...
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1answer
26 views

Inverse Function Theorem when determinant is undefined

For $f(x,y) = (x^3 - y^2, \sin{x} - \ln{y})$ f-inverse exists and is differentiable in a non-empty set around $(-1,0)$. Find $D(f^{-1})$ at $(-1,0)$. Seemingly this is an Inverse Function Theorem ...
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25 views

Inverse Image Proof

Let $f:X\rightarrow Y$. Let $A$, $A_1$ and $A_2$ be subsets of $X$ and $B$, $B_1$, and $B_2$ be subsets of $Y$. Then, I need to prove that $f^{-1}(B_1\cup B_2)=f^{-1}(B_1)\cup f^{-1}(B_2)$. I know ...
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2answers
29 views

What is needed to apply the inverse function theorem to $f(x,y,z) = \left(\frac{ax^2 + by^2}{2}, \frac{cy^2+dz^2}{2}, \frac{ex^2 + fz^2}{2} \right)$?

Let $f:\mathbb{R}^3 \to \mathbb{R}^3$ be $$f(x,y,z) = \left(\frac{ix^2 + hy^2}{2}, \frac{jy^2+kz^2}{2}, \frac{mx^2 + nz^2}{2} \right).$$ My question is what restrictions are necessary on ...
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1answer
46 views

Inverse of matrix mod $26$ wolframalpha wrong

I want to find $A^{-1} \pmod{26}$ for $A=\begin{bmatrix}10&3\\5&3\end{bmatrix}$ and I did the conventional $\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$ and found the ...
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1answer
1k views

Finding the derivatives of inverse functions at given point of c

Hoping someone can help me the understand the steps to solve a problem like this. I'm guessing it involves the formula: $\frac{d}{dx}f^{-1}(f(x))=1/f'(x)$. Am I right in this assumption? I would post ...
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2answers
24 views

Is there any way to test the existence of left or right inverse matrix?

I know that the inverse matrix of a square matrix exists iff its determinant isn't 0. What about a non-square matrix? Is there any theorem about the existence of a ...
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1answer
79 views

Prove $\frac1{\sqrt x}$ is continous on $(0,\infty)$. Stuck on last line!

Let $f(x) = \frac1{\sqrt x}$ for $x\in(0,\infty)$. Given $\varepsilon>0$ and $x_0\in(0,\infty)$, show there exists $\delta>0$ such that $$|x-x_0|<\delta$$ implies that $$|f(x)-f(x_0)| ...
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1answer
30 views

Finding the domain of a difficult inverse

$f(x)=\frac{3x+5}{-6x+2}$ , largest possible domain Find $f^{-1}(x)$ of this 1-1 function and the domain. So I wrote the equation as $$y=\frac{3x+5}{-6x+2}$$ Interchanged x and y, and made y ...
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3answers
319 views

how to find two right-inverse functions of a function

i am stuck in this problem. i need to find two right-inverse functions of this function: $h: \Bbb N_0\times \Bbb N \to \Bbb N, (m,n)\mapsto m+n$. i know that the function h' is a right inverse of ...
2
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2answers
195 views

Left inverse of a function

Let $f$ be the function $f\colon \mathbb{N}\rightarrow\mathbb{N}$, defined by rule $f(n)=n^2$. Needed to find two left inverse functions for $f$. I know only one: it's $g(n)=\sqrt{n}$. Does anyone ...
0
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2answers
44 views

Computation of determinant for Using Inverse Function Theorem

Let $f : \Bbb R^{3} \setminus \{(0, 0, 0)\} → \Bbb R^{3} \setminus \{(0, 0, 0)\}$ be given by $f(x, y, z) = (x/(x^{2} + y^{2} + z^{2}), y/(x^{2} + y^{2} + z^{2}), z/(x^{2} + y^{2} + z^{2}))$. Show ...