# Questions tagged [inverse]

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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### Method checking to prove the below statement regarding a function and its inverse

If a function "$f$" has a inverse and given that there is a point $x= a$ (lying in the domain of $f$) such that $f(a) = b$ , $a\neq b$ and $f^{-1}(a) = f(a)$ . Show that $f(x)$ is a self-...
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### Range of $\cot^{-1}(x)$

S. L. Loney mentions the range of $\cot^{-1}(x)$ to be $[-\pi/2,0)\cup(0,\pi/2]$ whereas my textbook (my coaching institute package) mentions it to be $(0,\pi)$. May I know which one is the correct ...
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### Solution to equation with moore-penrose inverse

I have a linear equation of the form $$C = (I-AA^+)X$$ where my variable is $X$ and $A$ is an hermitian operator and $A^+$ is the pseudo inverse. Assuming that the determinant of $A$ is $0$ (or ...
1 vote
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### Domain and range of $f(x)=3\sin^{-1}(5x)$

I am trying to determine the domain of $$f(x)=3\sin^{-1}(5x).$$ Since the domain of $g(x)=\sin^{-1}(x)$ is given by the set $\{x\in\mathbb{R}:-1\leq x\leq 1\}$, I understand that the domain of $f$ ...
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### Equality of product of matrices

I have been wondering if the following statement is corrrect? Assume that $A$ is $n \times m$ matrix where $m \leq n$ and rank of $A$ is $m$, $X$ and $B$ are $m \times m$ square matrix and it is ...
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### Matrix identity involving inverse

Let matrix $Y \in \mathbb{S}^{n}$ be symmetric and positive definite (thus invertible) and $X \in \mathbb{R}^{n \times n}$ such that $Y - X X^T \succ 0$. I think that the following identity should ...
1 vote
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### Is this an incorrect application of Tutte's theorem of perfect matching for bipartite graphs?

This is an extract from a conference paper. It seems the authors are invoking Tutte's theorem (since  refers to the 1947 paper) to conclude that a matrix $J(x)$ with given numerical entries is ...
1 vote
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### Suppose that $dim(U) = dim (V)<\infty$ and let $T\in Hom(U,V)$. If $AT=\iota_V$ or $TA=\iota_U$, then $A$ is an isomorphism and $A^{-1}=T$.

I have a question regarding the proof of this corollary 2.11, which I have encountered in a course on Linear Algebra. As this is a corollary, I have a feeling that the proof should be somewhat obvious ...
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### Inverse of Grassmann variables

Let $\theta$ be a Grassmann variable. I know that $1/\theta$ is not defined but $\frac{1}{1-\theta}=1+\theta$. My question is simple: is $\frac{1}{\bar{\theta}\theta}$ defined and if so how? My guess ...
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### Division (by 0) and multiplication (as its inverse operation).

Division (by 0) and multiplication (as its inverse operation). $\frac{a}{b} = c \implies a = bc.$ $b (\frac{a}{b}) = a.$ If this is true, why is an explanation for the reason you cannot divide by zero ...
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### Power series for $(A+\epsilon I)^{-1}$ for large $\epsilon$

I am trying to figure out a power series for $(A+\epsilon I)^{-1}$ when $A$ is an invertible matrix and $\epsilon$ is large. The Neumann series can be used when $\epsilon$ is small. Is there something ...