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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Find the inverted matrix of $A=(a_i,_j),a_{i,j}={\dbinom{j-1}{i-1}}$

Let be $A=(a_i,_j)\in M_{n+1}(\mathbb{R})$ defined for all $(i,j)\in [\![ 1,n+1 ]\!]^2$, by $a_{i,j}={\dbinom{j-1}{i-1}}$. Let's show this is invertible and determine its inverted matrix. To my ...
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1answer
44 views

Calculate $59x^{-1}\equiv 1 \pmod{63}$ [closed]

How can I calculate $$59x^{-1}\equiv 1\pmod{63}?$$ I only know that $59$ is prime.
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32 views

Asymptotic to $f^{-1}(f ' (x)) $?

Let $tr(n)$ be the triangular numbers and $te(n)$ be the tetrahedral numbers. $$g(x) := \sum \frac{x^n}{n! 2^{tr(n)}}$$ $g'(x) = g(\frac{x}{2}) $ Now consider the analogue $$ f(x) = \sum ...
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23 views

Sherman-Morrison formula and a sum of outer products

A specific form of the general Sherman-Morrison formula reads $(1+u v^T)^{-1}$ = $1 - \frac{u v^T}{1+v^T u}$ where $1$ is the identity matrix, $u,v$ are vectors (say with length n) and T denotes ...
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33 views

Classifying functions whose inverse do not have a closed form

My initial question contained about how to classify functions whose integrals and inverses do not have a closed form. But I found this question: How can you prove that a function has no closed form ...
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1answer
36 views

Proof of the Sherman-Morrison Formula

I was reading a few proofs for the Sherman-Morrison Formula, which states that if $A$ is invertible and $M = A + \mathbf{u}\mathbf{v}^T$, then $M^{-1}$ is given by: $$A^{-1} - A^{-1}\mathbf{u} ...
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4k views

Given $g(x)$ and $f(g(x))$, solve for $f(x)$.

I've hit a wall on the above question and was unable to find any online examples that also contain trig in $f(g(x))$. I'm sure I am missing something blatantly obvious but I can't quite get it. $$ ...
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1answer
25 views

Finding modular inverse (wrong approach)

I'm trying to find the modular inverse of $$30 \pmod{7} $$ I have tried using the Euclidean algorithm and it gave me the right answer, which is $x \equiv 6 \pmod{7} $. However, I tried using another ...
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53 views

Easy way to find an inverse in $Z_n$

Well, I'm solving for x in $11x=3$ in $Z_{12}$. And the way for me to do this is finding the inverse of 11 in $Z_{12}$. But to get the inverse, I've tried all possible elements in $Z_{12}$ so that ...
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2answers
36 views

Solving $3\times 3$ matrix equations:

I am familiar with finding the inverse of matrices, but struggle to formulate matrix equations. In this particular question, one is asked to find the elementary matrix E where $E*A = B$. $A$ is ...
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1answer
34 views

sum of matrix inverse problem

Recently, when I was reading matrix analysis, a formula confused me a lot: If $A+B$ is nonsingular, then the following is true, $$A(A + B)^{-1}B = B(A + B)^{-1}A$$ I tested some random ...
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1answer
34 views

Logarithmic to linear

Given this function: $$\frac{1.0}{1024.0} + \frac{x}{100.0} * \frac{1023.0}{1024.0} = y$$ $$10 * \frac{\log_{10}(y)}{\log_{10}(2)} = z$$ $$z * 100 = a$$ ...
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2answers
51 views

What kind of distribution in this chart?

Could you tell me what kind of distribution is this? Chart This is the data: ...
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1answer
53 views

Relationship between inverse of related matrices

Suppose I have a matrix $A \in \mathbb{R}^{m\times n}$ with $m \geq n$ and suppose that a matrix $G=(A^T A)^{-1}$ exists. Now suppose that I have an other matrix $B \in \mathbb{R}^{m\times m}$ that ...
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1answer
21 views

Is this system invertible?

$y(t) = \int\limits_{-\infty}^{\infty} \frac {x(t)^2}{x(t-1)} dt\\$ I was trying to prove or disprove the invertibility of this function. The only thing I could think of was differentiating it. But ...
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2answers
45 views

Compute the indicated power of a matrix

Compute the indicated power of the matrix: $A^8$ $ A = \begin{bmatrix}2&1&2\\2&1&2\\2&1&2\end{bmatrix} $ I calculated the eigenvalues: $ \lambda_1 = \lambda_2 = 0, \lambda_3 ...
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2answers
47 views

Is this kernel invertible ? $K(x,y)=\frac{e^{-\frac{xy}{x+y}}}{x+y}$

Is the following Kernel invertible? $K(x,y)=\frac{e^{-\frac{xy}{x+y}}}{x+y}, x\in[0,1],y\in [0, \infty)$ i.e. if $\int_0^1 K(x,y) f(x) dx=0 ,\forall y\in [0, \infty)$ can we conclude $f(x)=0,x\in ...
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0answers
31 views

Confusion regarding logic in paper, “A NOTE ON THE INVERSION OF POWER SERIES,” published in the AMS journal

I was reading "A NOTE ON THE INVERSION OF POWER SERIES" and was able to follow the paper's reasoning until the bottom of the second page, where it states: in fact we can calculate the power series ...
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1answer
23 views

Trace of Hermitian Positive Semidefinite Matrix

Well, the question I want to ask is as follows. Suppose A and B are Hermitian Positive Semidefinite (PSD) matrices, I wonder if it is possible to prove $Tr(A*(A+B)^{-1})\in (0,1]$ (if it is ...
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1answer
49 views

Finding inverse polynomial in $\mathbb{F}_p[x]/(\psi)$ with maple

I need help with maple. I want to invert the polynomials $(x^{361}-x)^2$ and $4(x^3+2x+1)^{19}$ with the help of maple. The problem is that we are working in $\mathbb{F}_{19}[X]$ and modulo ...
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44 views

'Stable' Ways To Invert A Matrix

So lets say that I need to invert a matrix that is generally dense and is poorly conditioned. What are some ways I can get an accurate inverse? Here are my candidates: SVD Inverse Inverse Via ...
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2answers
64 views

If a matrix $A^2$ is invertible, is $A^3$ invertible? [closed]

I know how to find out if a matrix $A^2$ is invertible if $A^3$ is invertible, but how can you find out invertibility if it's the the other way around?
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50 views

Solving for $x$ in a Laplace equation

So I have this Laplace equation: $$s^{2}x+4sx+5=\frac{s}{s-1}$$ And I want to solve for $x$. My result is the following: $$x = \frac{5-4s}{s^{3}+3s^{2}-4s}$$ Which is also the same answer that for ...
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2answers
65 views

Finding all left inverses of a matrix

I have to find all left inverses of a matrix $A = \begin{bmatrix} 2&-1 \\ 5 & 3\\ -2& 1 \end{bmatrix}$ I created a matrix to the left of $A$, $\begin{bmatrix} a &b &c \\ ...
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3answers
65 views

Necessary and/or sufficient conditions for $A+B$ to be invertible

Let $A$ and $B$ be two $n\times n$ real invertible matrices. Are there necessary and/or sufficient conditions (involving only $A$ and $B$ separately, not $(A+B)$ iteself) for $A+B$ to be invertible? ...
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2answers
30 views

Inverse of the Cross Ratio for Mobius Transformation from Circle to Circle

I'm reading Conway's complex functions of one variable, and in chapter 3 he goes over Cross-Ratios. He defines the cross ratio to be $(z,z_1,z_2,z_3)=\frac{(z-z_3)(z_2-z_4)}{(z-z_4)(z_2-z_3)}$, where ...
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2answers
38 views

What does it mean for f([x])=[2x] for a function mapping R/Z to R/Z?

Let X=R/Z (the circle), with a map $f : X → X$ given by $f([x]) = [2x]$. I'm a little lost on what $f([x]) = [2x]$ means. I thought the function was mapping the equivalence class [x] to the ...
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1answer
32 views

Linear Algebra - Real Matrix and Invertibility [closed]

Let $M=\begin{pmatrix}A&B\\C&D\end{pmatrix}$ be a real matrix $2n\times 2n$ with $A,B,C,D$ real matrices $n\times n$ that are commutative to each other. Show that $M$ is invertible if and only ...
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55 views

how to prove that invertible matrix and vectors span the same space?

Given $M$ is an invertible matrix, and {$\vec{v_1}...\vec{v_k}$} spans $R^n$, then {A$\vec{v_1}...A\vec{v_k}$} also spans $R^n$ What does matrix invertibility have to do with span?
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1answer
52 views

Inverse of linear combination of trigonometric functions [closed]

I have an equation of the form: $$\tan(y)=\alpha_1\cos(x)+\alpha_2\sin(x)$$ where $x$ and $y$ are in $(0,2\pi)$ and the coefficients are real numbers. Implicitly this defines $y$ as a function of ...
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44 views

Orthogonal matrix problem

So the question asks: Let $A$ and $B$ be n×nn×n orthogonal matrices, with $n≥2$. Which of the following matrices must be orthogonal? A. $A^TB$ B. The matrix C obtained by multiplying the second ...
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1answer
37 views

Show that Ax = b is solvable when [A b] is singular.

I have the following problem: Review: Suppose A is 5 by 4 with rank 4. Show that Ax = b has no solution when, the 5 by 5 matrix [A b] is invertible. Show that Ax = b is solvable when [A b] is ...
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2answers
35 views

Invertible function that “messes” order [closed]

I am looking for an invertible discrete function $f$ such that given some integer n, if i apply $f(i)$ for $i=0,\dots,n$ I would get all the integers in range $[0..n)$ exactly once, but in a "messy" - ...
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27 views

Inverse Functions (Discrete Math)

Say you have $f: \mathbb{Z} \to \mathbb{Z}$ defined by $f(x,y) = (2x+y, y)$ How would you check if the function was invertible? As well as determining it's inverse if it is? Thank you
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2answers
24 views

How can I invert the asymptotic form $x^{3/2}=y^{3/2}(1+a/y^2 + … )$ to find $y=y(x)$?

This might sound silly, but the fact there's a $a/y^2$ term in the expansion made me feel a little lost. Could anyone help? Thanks
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1answer
21 views

When left inverse of a function is injective

Consider function $f^{-1}$ which is a left inverse of another function $f$. I require that $f^{-1}$ must be injective. What does it tell me about $f$? In other words, can I put some constraints on $f$ ...
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2answers
27 views

Need two functions always be composed to prove they are inverses?

Normally, if I claimed that $f: A \rightarrow B$ and $g: B \rightarrow A$ were inverses of each other, I would check for the following results: $f \circ g(b) = b$ and $g \circ f(a) = a$. Suppose I ...
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0answers
22 views

Calculating the inverse with variables that include logarithm and don't.

I am trying to calculate the inverse of this function and failing. $y_1 =z_1 \sqrt\frac{-2* log(z_1^2 + z_2^2)}{(z_1^2 + z_2^2)}$ Is there a systematic way to go about it?
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20 views

Matrix computation of products

If I have two $n \times n$ matrices $A$ and $B$ and a vector $c$,how would I compute the product of $A^{-1}Bc$ ? I know how how to get $A^{-1}$ by doing LU decomposition of $A$ but how do I translate ...
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2answers
60 views

Inverse of matrices that differ only by one element

Let's suppose that a real matrix $\textbf{A}_{n\times n}$ is nonsingular and its inverse is $\textbf{A}^{-1}_{n\times n}$. Next we change its $A_{ij}$ element to $A_{ij}+a$ and we keep all the other ...
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2answers
36 views

Finding the inverse of a matrix given an equation

So I've been given this equation: $A\begin{bmatrix} 2&3&1&5\\ 1&0&3&1\\ 0&2&-3&2\\ 0&2&3&1 \end{bmatrix} = \begin{bmatrix} 0&1&0&0\\ ...
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4answers
65 views

What is the difference between $f(x)=x^2 +1$ and $f(x)=x^3 -1$ when finding the inverse?

I'm doing some exercises on computing the inverse of each function. In exercise number 56 I did an example where I have to compute the inverse of the function. With my understanding $f(x)=x^2 +1$ ...
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2answers
45 views

Matrix invertibility proof? [closed]

Can it be proven that $A^\top A$ is invertible given just the fact that: if $A^\top Ay = \theta$ then $Ay = \theta$? Here $y$ is a vector and $\theta$ is the vector zero.
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1answer
54 views

inverse of $2\times2$ block matrix

I want to compute the inverse of the $2\times2$ block matrix $$ \left(\begin{array}{cc} A & P\\ P^T & 0\\ \end{array}\right), $$ with $A$ an $n\times{n}$ matrix and $P$ and $n\times{m}$ ...
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2answers
41 views

Prove that $(G,*)$ is a group.

Let $G$ be a non-empty set, $*:G\times{G}\to{G}$ a binary operation that satisfies: $*$ is associative. Exist $e\in{G}$ such that $a*e=a$ $\forall{a\in{G}}$ For all $a\in{G}$ there is ...
2
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1answer
59 views

What is the derivative of $\mathrm{trace}((S^T S)^{-2})$ w.r.t. $S$

I would like to compute the derivative of $\mathrm{trace}((S^T S)^{-2})$ w.r.t. $S$. I know that $\frac{\partial \mathrm{trace}((S^T S)^{-1})}{\partial S} = -2S(S^T S)^{-2}$ but I have a higher order ...
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3answers
2k views

Does the sum of the inverses of the sums of the primes converge?

$$\sum_{m=0}^∞ \frac{1}{\sum_{n=0}^m p_n} = \frac{1}{2} + \frac{1}{5} + \frac{1}{10} + \frac{1}{17} ... $$ Where $p_n$ is the $n$th prime number, does $\sum_{m=0}^∞ \frac{1}{\sum_{n=0}^m p_n}$ ...
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3answers
27 views

Inverse Function with Fraction.

I'm having an issue with this problem for solving for inverse function: $$f(x) = \frac{9x + 5}{x + 4}$$ Step 1: f(x) to Y. Then, change "$x$" to "$y$" in all cases. $$f(x) = \frac{9y + 5}{y + 4}$$ ...
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1answer
27 views

Inverse of a matrix whose elements are arrays

I have a group of data as in the following figure: $$A=\left[\begin{array}{ccc} [0.9\,\,0.6\,\,0.9\,\,0.2] & [0.4\,\,0.3\,\,0.1\,\,0.1] & [0.1\,\,0.3\,\,0.5\,\,0.6] \\ ...
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0answers
45 views

Explicitly understanding the implicit function theorem

Suppose I have a curve $f$ in $\mathbb{R}^2$, the implicit function theorem guarantees the existence of a smooth local inverse of this function $f$. Question: My question is is there a way to ...