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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If the function $f(x)=ax+b$ has its own inverse,then the ordered pair $(a,b)$ can be

If the function $f(x)=ax+b$ has its own inverse,then the ordered pair $(a,b)$ can be $(A)(1,0)\hspace{1cm}(B)(-1,0)\hspace{1cm}(C)(-1,1)\hspace{1cm}(D)(1,1)$ This is a more than one options correct ...
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Prove that $\frac{1}{[Z^{-1}]_{kk}}=\frac{\text{det}Z} {\text{det}Z_{kk}}=\text{det}Z_{kk}^{\text{SC}}$, $Z_{kk}^{\text{SC}}$ is the Schur complement

Suppose $Z$ is a complex (Wishart) matrix. Let $a=\frac{1}{[Z^{-1}]_{kk}}$, where $Z^{-1}$ is the inverse of $Z$ and $[Z^{-1}]_{kk}$ represents the $(k,k)$-th entry of $Z^{-1}$. When I was reading ...
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Finding the value of Inverse Trigonometric functions beyond their Real Domain

I wanted to ask how can we calculate the values of the inverse of trigonometric functions beyond their domain of definition, for example $\arcsin{2}$ beyond its domain of ...
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35 views

Can we always for an invertible matrix $M$ find real number $\alpha \neq 0$ such that $M+\alpha$ is invertible?

I do not know enough about matrices, maybe only enough to be able to create question like this one, but I would like to see an answer. Let $a_{ij}$ be some element of invertible $n\times n$ matrix ...
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28 views

Matrix Inversion distribution

How do you distribute the inversion in $(A^TA+\lambda I)^{-1}A^Ty$ assuming $A$ is a $n \times n$ square invertible matrix, $y$ is a vector with the dimension of $n$, and $\lambda$ is a constant?
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Implicit function theorem, what is the meaning of invertible linear operator?

I have to show that around $(1,-1,0)$(have to find the neighborhood as well), $x,y$ are determined uniquely by $z$ given $x+yz-z^3=1, x^3-xz+y^3=0$. What I did so far is: $f:\Bbb{R}^2\times\Bbb{R}\to ...
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12 views

Relation of numerical stability of matrix inversion and it's determinant

I have been taught that "inverting a square matrix with small determinant is numerically unstable because it is close to singular"? Is this right opinion?
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26 views

b is the inverse of a $( \mod 11)$

Let a and b be numbers in the set $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8 , 9, 10\}$ such that b is the inverse of a $(\mod11)$ and a and b are not equal. How many such subsets $ \{a, b\}$ of S are there?
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Production Model x=Cx+d — Use Inverse Matrix

Question: Consider the production model x = Cx + d for an economy with two sectors, where C= 0.0 0.5 0.6 0.2 and d= 50 30 Use an inverse matrix to determine the production level ...
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How to Prove that this Function is not $1-1$

I'm trying to show that the function $$f(x)= \dfrac{x}{4}+x^2\sin\left(\dfrac{1}{x}\right)$$ is not $1-1$ for any neighborhood of $0$. I know that what I have to do is find two different points that ...
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63 views

How can I determine B-inverse from an optimal tableau of a LP?

(This is NOT a homework question, I am reviewing for my upcoming exam) Given this linear program: and this optimal tableau: I am attempting to determine $B$ inverse using the table above. From ...
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74 views

Inverse of generalized arrow matrix $A = M^T * M + I$

If we have the following linear system: Ax=b And matrix A is created by multiplying a rectangular matrix with it's transpose: $A = M^T * M + I$ What is the best method to solve for x for different b ...
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2answers
31 views

Inverse of an exponential function

I am having difficulties forming the inverse of this $f(x) = 3 \cdot2^{3x+1} \cdot 5^{3x-1}$. What I have done so far: $3 \cdot 2^{3y} \cdot 2^1 \cdot 5^{3y}\cdot5^{-1} \Leftrightarrow 3\cdot 2\cdot ...
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1answer
26 views

Does an $x \in \mathbb Z$ with $xa\equiv_nb$ exist, if $gcd(a,n) $ divides $b$?

Does an $x \in \mathbb Z$ with $xa\equiv_nb$ exist, if $gcd(a,n) $ divides $b$ ? My idea: \begin{aligned} & xa &\equiv_n& \quad b \cr \Leftrightarrow \quad& x& \equiv_n& \quad ...
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3answers
47 views

Is $f(x) = e^x$ one-to-one if $f:\mathbb{R} \rightarrow \mathbb{R}$?

My book says that $f(x) = e^x$ is not invertible from the set of real numbers to the set of real numbers. But I disagree since $f(x) = e^x$ is injective with this given domain and codomain and ...
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24 views

Inverting a sparse Matrix

I have a sparse, square, symmetric matrix with the following structure: (Let's say the size of the matrix is N x N) the structure of the sparse matrix Here, the area under the blue stripes is the ...
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16 views

Proof of the inverse function theorem in van der Vaart

I have a question regarding the proog of lemma 4.3 in van der Vaart at p.36 ...
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29 views

Proving inverse function theorem

I'm asking for an help to understand the proof of the inverse function theorem, in particular one part of it represented by Lemma 4.2 in van der vaart p.36 (you can find it here ...
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1answer
22 views

Inverting this function

I have to invert this function: $$f: \mathbb{N} \rightarrow \mathbb{N}, f(n) = \begin{cases} n+1, & \text{if $n$ is odd} \\ n-1, & \text{if $n$ is even} \end{cases}$$ But I am not able to ...
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1answer
32 views

Inverse Laplace Transform of $\frac{1}{\sqrt{s+a}+\sqrt{s+b}}$

I need to calculate the inverse laplace of: $$F(s)=[\frac{1}{\sqrt{s+a}+\sqrt{s+b}}] \qquad \qquad (s>-a\quad ;\quad s>-b;\quad a\neq b) $$
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49 views

Show that $P$ is symmetric.

Let $P$ = $A(A^TA)^{-1}A^T$, where A is an m x n matrix with rank $n$. I feel like this is wrong, but here is my attempt: $A(A^TA)^{-1}A^T$ = $AA^{-1}(A^T)^{-1}A^T$ = $I$ And $I^T$ = $I$, so the ...
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42 views

Show that $P^2$ = $P$

Let $P$ = $A(A^TA)^{-1}A^T$, where A is $m \times n $ 0f rank $n$. This is the projection matrix, right? Every site I've been on says that this is the projection matrix such that $P^2$ = $P$, but ...
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48 views

Inverse Laplace Transform and error function

Express your answer in terms of the error function: $$L^{-1}\left[\frac{1}{\sqrt{s^3+as^2}}\right]$$ Clue: $\qquad L\left[\frac{1}{\sqrt{t}}\right]=\sqrt\frac{π}{s} \qquad , \qquad s>0$ Error ...
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2answers
37 views

Getting the derivative of the inverse of a function

Given $f(x)$, how would I find $(f^{-1})'(x)$? As an example how would I find that for this problem: $f(x) = 4x^3 + 5x + 2$
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1answer
23 views

Limits, Determinants and Inversion of a matrix-valued function

Suppose I have a matrix-valued, continuous function $$A\colon [0,\infty) \to \mathbb R^{n\times n},\qquad h\mapsto A(h).$$ I know that for the limit $h\to 0$ the matrix is invertible: ...
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38 views

How do you find inverse for certain exponential function?

How do you find inverse for $y=\frac{e^x-e^{-x}}{2}$?
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27 views

Prove a function with a positive definite Jacobian is 1-1

Suppose that $f:\mathcal{O}\subset\Bbb{R}^n\to\Bbb{R}^n, \mathcal{O}$ open and convex, $f\in C^1(\mathcal{O})$, and the symmetric part of the Jacobian matrix, $\frac{(Df)(x)+(Df)^T(x)}{2}$, is ...
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53 views

a practical question about matrix derivative with inverse and chain rule: dimension mismatch

Recently, I was trying to take the following derivative $$ \dfrac{\partial (X^TV^{-1}X)^{-1}}{\partial V} $$ I was referring to matrix cookbook to solve it, where I found several useful equations: ...
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110 views

Why does Arccos(Sin(x)) look like this??

I can kind of understand the main direction (slope) of $y$ over the different $x$ intervals, but I can't figure out why the values of $y$ take on the shape of straight lines and not curves looking ...
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25 views

Consider $f : \mathbb{N} → \mathbb{Z}$ defined as $f (n) = \frac{(−1)^n (2n−1)+1}{4}$. Find its inverse.

I cannot find an inverse of this function for f(n) = x, where x is an integer, that gives out a natural number. Some guidance would be very helpful... I already know the function is bijective so there ...
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21 views

Hessian for inverse probit link

I'm trying to calculate Hessian and Fisher Information for binomial model using inverse probit link, Suppose likelihood function is $L(\pi)=\prod\limits_{i=1}^n \pi_i^{y_i}(1-\pi_i)^{1-y_i}$ and ...
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3answers
33 views

For which values of is the following matrix invertible and what is its inverse?

For which values of is the following matrix invertible and what is its inverse? $$A =\begin{bmatrix}1 & 1 &0\\1 & 2 & 2\\1 & 2 & \lambda\end{bmatrix}$$ If someone can please ...
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4answers
39 views

Inverse of function - Difficulty solving [closed]

$$y=\sqrt{35\tan (\frac{\pi }{180}x)}$$ I have a really hard time finding the inverse of this particular function. Can anyone shine some light on why that may be, or alternatively solve it if I'm ...
2
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1answer
40 views

Proving a matrix is invertible

There's a linear algebra problem I'm having some trouble with: Let $A$ and $B$ be square matrices with the dimensions $n\times n$. Prove or disprove: If $A^2 + BA$ is invertible, then $A$ is also ...
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Are these equiv? $\cot^{-1}(-x) = -\cot^{-1}(x)$

$-\cot(y)=x$ Let's say you want to put this in terms of y... $\cot(y)=-x$ $y = \cot^{-1}(-x)$ =================== Is this also valid ? $-\cot(y)=x$ $y = -\cot^{-1}(x)$ If yes, how exactly does ...
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Is the inverse of f(x) considered a function [closed]

In a general, is the inverse of function considered a function?
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49 views

If the product of two square matrices is invertible, then both matrices are invertible

If $A$ and $B$ are $n\times n$ matrices, and $AB$ is invertible then $A$ and $B$ are invertible. I started out by writing that since $AB$ is invertible, then for the equation $ABx=b$ has a ...
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2answers
33 views

If f(x+y)=f(x)*f(y) and f is a bijection, show that its inverse satisfies this function equation

I'm having trouble with this problem. I'm not even sure how to go about finding the inverse of an equation with both x and y. Here is the problem: If $f(x+y)=f(x)*f(y)$ and $f$ is a bijection, show ...
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6answers
35 views

The multiplicative conjugate of an invertible matrix is invertible

If $A,B,C$ are $n \times n$ (real) matrices and $A$ and $B$ are invertible, with $AB=BC$, prove that $C$ is also invertible. My attempted proof is $(B^{-1})(AB) = (B^{-1})(BC)$. Then ...
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1answer
29 views

Properties of adjoint matrix in a finite dimensional inner product space

let $V$ be a finite dimensional inner product space. Let $T$ be a linear operator on $V$. Prove that there exists an invertible linear operator $U$ such that $U^{-1}TT^*U = T^*T $ where $T^*$ is ...
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why inverse trigonometric function DNE

Determine the exact value $\arccos\left[\sec\left(\dfrac{7\pi}{6}\right)\right]$ and $\text{arcsec}\left[\sin\left(\dfrac{13\pi}{6}\right)\right]$ Why does the exact value of these two questions not ...
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61 views

How to find the second derivative of the inverse function of $f(x)$ at $x=0$?

If $f\displaystyle(x)=\int _{ \sin x }^{ \cos x }{ \frac { dt }{ e^{ t }\sqrt { 1-t^2 } } } $ where $x\in[0,\pi/2]$ then how to find the second derivative of the inverse function of $f(x)$ at $x=0$ ...
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Calculating differential of inverse function.

trying to find $(f'^{-1})(a)$ and am getting the wrong answer.
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1answer
45 views

Finding an inverse matrix

I should find the inverse matrix of the matrix: \begin{pmatrix} n & n & \cdots & n & n \\ n-1 & n-1 & \cdots & n-1 & 0 \\ \vdots & & & & ...
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1answer
17 views

fine the inverse of $[2]$ and $[23]$ in$ \mathbb{Z}_{41}$

I know the inverse of [23] is [23] * [25] = 575 575 congruent to 1 mod 41 [25] is the inverse I have started the other one but I am doing something wrong I got [2] * [41] = [82] = [0] 82 ...
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42 views

If $f:\{1,2,3\}\to\{1,2,3\}$ is bijective and $f(1)=2$, can we verify that $f(2)=1$?

I know that if $f$ is bijective, if $f(1)=2$ then $f^{-1}(2)=1$ but if $f:\{1,2,3\}\rightarrow\{1,2,3\}$ then does it mean necessarily that $f(2)=1$?
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37 views

proof: inverse of lower triangular identity matrix

As you know that is enough negating below of diagonal to inverse of lower triangular identity matrix. example $$A = \left(\begin{matrix} 1 & 0 & 0 & 0 \\ 3 & 1 & ...
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44 views

The column space of $A^2$ is all of $\mathbb R^n$ if and only if the column space of $A$ is all of $\mathbb R^n$

How would I go about proving the following statement? "Let $A$ be an $n \times n$ matrix. $\operatorname{Col}(A^2)=\mathbb{R}^n$ if and only if $\operatorname{Col}(A)=\mathbb{R}^n$" I started off by ...
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29 views

how do I find the inverse function of $x = ky - \cos(y)$ where $k$ is a constant

In other words, I'd like to express $y$ in terms of $x$ instead of having $x$ expressed in terms of $y$. I just don't know how to proceed!
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2answers
34 views

Square of Elementary Matrix Proof

I'm having trouble proving the following statement: "There exists an elementary matrix $E_1$ such that $E_1^2 = I$" I'm thinking about how the inverse of $E_1$ is equal to $E_1$ (so $E_1^{-1} = ...