# 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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### What are all the Banach algebras where $\|a\|\|a^{-1}\|=1$? [on hold]

Is there a characterization for all Banach algebras such that $\|a\|\|a^{-1}\|=1$ whenever $a$ is invertible?
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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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### Inverse of $f(x) = 2x^2+8x+13?$

How can you find the inverse of $f(x) = 2x^2+8x+13?$ This is what I've tried so far: $y = 2x^2+8x+13$ $x = 2y^2+8y+13$ $x-13 = 2y^2+8y$ $x-13=y(y+8)$ This is where I got stuck. To be clear, I want ...
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### Time complexity of inverting an $n \times n$ matrix which is the sum of a rank-$m$ matrix and a full-rank diagonal matrix

I want to know the time complexity of inverting $K$, where $K$ is an positive-definite $n\times n$ matrix: $$K=\Lambda+Q$$, where $\Lambda$ and $Q$ are both $n\times n$ matrix, $\Lambda$ is a full-...
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### Inverse image of a function in multivariable calculus?

Let $f: R^2 \rightarrow R^2$ defined by $f(x,y) = (x+y,xy).$ Claim : Inverse image of each point in $R^2$ under f has at most two elements. My Claim : Suppose $f(x,y) = (x+y,xy)= (p,q).$ We have ...
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### value of an Inverse trigonometry expression

How can i find the value of $\alpha = \arcsin\frac{\sqrt{63}}{8}$ to substitute in the expression , to value of $\sin^2(\frac\alpha4)$ ?
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### Inverse Laplace transform of $(s^2-1)^{-1/2}$

please help with this. Not derived from any differential equation. Also found the answer $\mathcal{L}^{-1}(\dfrac{1}{\sqrt{s^2-1}})$
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### Continuous dependence of matrix elements

I've stumbled upon several solution of linear algebra problems which use notion of "continuous dependence" of matrix polynomials on matrix elements. For instance (translated, so any inaccuracies are ...
0answers
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### Math notation to show two numbers in a range that added together get the max of the range [closed]

I am completely new to math notations, it's been about 30 years since high school, and I am writing a research paper (completely on my own, not for a degree). I basically want to show that two real ...
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### Alternative view of matrix inversion (explanation required)

We were taught in linear algebra that in order to try to find the inverse of a matrix we can create an augmented matrix $[AI]$ where $A$ is the original matrix and $I$ is the identity matrix. Then we ...
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### Relation between powers of inverse modulo n.

Recently, I was studying enchanced euclidean algorithm. I am wondering if there is some way to calculate inverse of $a^2$ (and higher powers) modulo $n$, knowing inverse of $a$ modulo $n$. For example:...
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### Value of the given expression …

If $$y=\tan^{-1}\left(\sqrt{\dfrac{1+\cos x}{1-\cos x}}\right)$$ then value of $(2x+14y)^3-343$ is ? I reduced the equation as $y=\tan^{-1}\left(\dfrac{1+\cos x}{\sin x}\right)$ but I couldn't ...
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### Using inverse of transpose matrix to cancel out terms?

I am trying to solve the matrix equation $A = B^TC$ for $C$, where $A$, $B$, and $C$ are all non-square matrices. I know that I need to utilize $M^TM$ in order to take the inverse. I'm just not sure ...
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### Function inversion (analytical)

Can $t(x)$ be found from: $$A \, t + B\ln\frac{1-t}{t}=x \; ?$$ Here, $A>0, \; B < 0$ and $0 \lt t \lt 1$. The $t(x)$ should be given in analytical form (even if you use, say, Lambert's W - ...
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### Using the Affine cipher, do we need $a^{-1}$ if we know gcd(a,26)=1?

I have just attempted the affine cipher with the word "code" $CODE = 02140304$ Lets choose our key as $(5,3)$, so our encryption is $y=5x+3$ $13211823=NVSX$ Now, to undo the code, I would have to ...
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### Can $A^{T}(AA^{T})^{-1}A$ be simplified?

Let $A$ is an $m\times n$ ($m<n$) real matrix with full positive entries and $\text{Rank}(A)=m$. Thus $(AA^{T})^{-1}$ is an $m\times m$ symmetric $M$-matrix since $AA^{T}$ is nonnegtive and ...
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### When A and B are of different order given the $\det(AB)$,then calculate $\det(BA)$

Let 'A' be a $2 \times 3$ matrix where as B be a $3 \times 2$ matrix if $\det(AB) = 4$ the find value of the $\det(BA)$ My attempt: I took A =  \begin{bmatrix} 2 & 0 &0\\ ...