# Tagged Questions

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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### Row sum of inverse of a matrix

Let's say I have a matrix A, $$A= \begin{bmatrix} a_{11}& a_{12} & a_{13} \\ a_{21}& a_{22} & a_{23} \\ a_{31}& a_{32} & a_{33} \end{bmatrix}$$ All the elements of A are ...
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### Questions about matrix rank, trace, and invertibility

(a) Prove that a square matrix $T$ of rank one has $\text{tr}(T)=0$ if and only if $T^2=0$. (b) Consider a matrix $A$ of the form $A=aI+T$, where $a\ne0$, $I$ is the identity matrix, and $T$ ...
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### Inverse of binary matrix

I have tried creating an inverse of a binary matrix using the identity matrix method. Have an identity matrix alongside the square matrix and perform all the operations to convert the square matrix to ...
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### Square Root of a matrix: transpose or inverse of eigen vectors?

Here is described that the square root of a matrix is defined as K^1/2 = V*D^1/2*V^-1 At the end of scetion 4 of this paper we can see W = K^-1/2es In the ...
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### How to obtain the inverse of $MSM^T$ when $(MM^T)^{-1}$ is already known and $S$ is an invertible symmetric matrix?

How to obtain the inverse of $MSM^T$ when $(MM^T)^{-1}$ is already known and $S$ is an invertible symmetric matrix? Assume that $M$ is an $n \times m$ matrix with $n \leq m$. Is it possible to obtain ...
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### Is there a simple algorithm to compute polynomial inverses over cyclotomic polynomials?

I'm working with polynomial inversions in a ring built over the nth-cyclotomic polynomial, with $n = 2^i$. As usual, I'm applying Extended Euclidean algorithm on this, an approach that does not scales ...
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### $y=x/(1+a(x))$, $\quad$ $x=y/(1+b(y))$. What is known about $a\mapsto b$?

\begin{align} y & = f(x) = \frac x {\displaystyle 1 + \sum_{n=1}^\infty a_n \frac{x^n}{n!}} \\[15pt] x & = f^{-1}(y) = \frac y {\displaystyle 1 + \sum_{n=1}^\infty b_n \frac{y^n}{n!}} \end{...
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### Is this how eigenvalues of some matrix $A$ are related to the inverse of $A$?

Let $A$ be an invertible $n\times n$ matrix. If $$Av = \lambda v \qquad (1)$$ for some $v$ and $\lambda$ then $\lambda$ is an eigenvalue of $A$ and $v$ a corresponding eigenvector. Equation $(1)$ may ...
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### Inverse operation of tetration and how it is computed?

If $c=a+b$, then $a=c-b$ and $b=c-a$. If $c=a\times b$, then $a=\frac{c}{b}$ and $b=\frac{c}{a}$. If $c=a^b$, then $a = \sqrt [b]{c} =c^{\frac{1}{b}}$ and $b=log_ac$. What are the analogous inverse ...
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### Is $\text{arccosec}(x) = \arcsin\left(\frac{1}{x}\right)$ for all $x \in ℝ?$

Is $\text{arccosec}(x) = \arcsin\left(\frac{1}{x}\right)$ for all $x \in ℝ?$ I'm still really new to trigonometric inverses, so if the above was cleared up I'd be grateful. Thanks.
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### Complex inverse function

I've got a problem when solving an inverse function. Usually when I have a basic function and trying to find its inverse is not a problem. I just solve for X and find it. But now I've got a more ...