# 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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### Inverse Laplace transform of $\exp(-s^4)$ [closed]

Can anyone help me in finding the Inverse Laplace transform of $\exp(-s^4)$ Thank you
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I have a $n\times m$ matrix, $C$, and a $m\times m$ matrix, $B$. Is there any computational trick that allows me to calculate $(CBC^T)^{-1}$ without first calculating $Z=CBC^T$? Specifically, when $m&... 0answers 50 views ### How to inverse the laplace transform$\frac{1}{\cosh(5\sqrt{s})}$? Let$X$be a random variable with$ E[e^{-sX}]=\frac{1}{\cosh(5\sqrt{s})} $and density function$f$. How to give a formula for$f$? 2answers 67 views ### Does every mathematical operation has an inverse operation? For example, we say that the addition and subtraction are inverse operations like that does each and every mathematical operation has an inverse operation? 1answer 43 views ### How to invert$f(x)=\dfrac{2^x}{1+2^x}$[closed] I know that if $$f(x)=\frac{2^x}{1+2^x}$$ then $$f^{-1}(x)=\log_2 \frac{x}{1-x}$$ How can I show this? 1answer 66 views ### Approximating the inverse of a polynomial function for camera calibration The method below to remove lens distortion from a camera was written more than ten years ago and i am trying to understand how the approximation works. ... 1answer 41 views ### Find the inverse of the function:$f(x)=x^9+x$Let$f(x)=x^9+x$. Show that$f$has an inverse and find the inverse. I don't seem to be able to find a way to start tackling this equation. Appreciate any tips on this question. 0answers 15 views ### Calculate inverse function Is there a way to calculate the inverse of the following function? $$s(t) =k\cdot \frac {e^{\alpha t} - 1}{e^{\beta t} - 1}$$ 2answers 37 views ### How do I proof that$A=\sum\limits_{i=1}^{m}x_{i}x_{i}^{T} $is invertible if and only if$X$has full rank? Show that$A=\sum\limits_{i=1}^mx_ix_i^T$is invertible if and only if$x_1,\cdots,x_m$span$\mathbb R^d$for$x_i\in\mathbb R^d$. Here are my thoughts: If$A$is invertible$Aw=0$only has the ... 3answers 2k views ### What is the most efficient way to find the inverse of large matrix? Let$A$be a large square$(n+1) \times (n+1)$invertible matrix, where$n \approx 1000$. $$A = \begin{bmatrix} -1 & 0 & 0 &\cdots & 0 & a_0\\ 1 & -1 & 0 &\cdots & ... 1answer 65 views ### What must be true for Ax = b to imply that x = bA^{-1}? What must be true for Ax = b to imply that x = bA^{-1}? Assume that A is a matrix, and x and b are column vectors. 1) Is A singular or nonsingular? From the given (implied) equation, it ... 1answer 48 views ### What is the shortest way to find inverse of a matrix? I know two methods to find the inverse of a matrix already:- Row and Column transformations A^{-1}= \frac{Adj(A)}{Det(A)} I want to know if there's any shorter method to do so because these two ... 1answer 79 views ### Invertibility of the matrix whose elements are the cube of the distance of the indices. I would like to prove, for any integer n>1, the invertibility of the n\times n matrix A whose elements are given by A_{ij}=|i-j|^3, where i and j are the indices. To be clearer, for ... 2answers 45 views ### Evaluate |A – |A|\operatorname{adj}A| if |A + |A|\operatorname{adj}A| = 0 where A is a non-singular matrix of order 2 A is a non-singular square matrix of order 2 such that$$|A + |A|\operatorname{adj}A| = 0$$where \operatorname{adj}A represents adjoint of matrix A, and |A| represents \det(A) . ... 1answer 47 views ### Block identity matrix inversion Let us consider the matrix A \in \mathbb R^{2N \times 2N} defined as \begin{equation} A = \begin{pmatrix} I & I+\Lambda_{12} \\ I + \Lambda_{21} & I\end{pmatrix}, \end{equation} where I ... 1answer 35 views ### Matrix relation for E_{ij}(a)=I_n+ae_{ij} I want to show that for pairwise differen i,j,k and the matrix E_{ij}(a) = I_n + ae_{ij}\in \text{SL}_n(K) the following relation holds:$$E_{ij}(ab) = E_{ik}(a)E_{kj}(b)E_{ik}(a)^{-1}E_{kj}(b)^{-... 1answer 57 views ### Showing a bound on$x^\intercal A^{-1} x$using$A$Let$A \in \mathbb{R}^{n \times n}$be symmetric and positive definite and fix vector$x_0 \in S^{n-1}$, which means$\|x_0\|_2 = 1$. We want to show $$x_0^\intercal A^{-1} x_0 \leq c$$ for a ... 1answer 24 views ### Inverse of DiracDelta at 0 is 99/5? When using Mathematica I've found an interesting result. InverseFunction[DiracDelta] == 99/5 (* returns True *) Or the inverse function of the DiracDelta ... 2answers 91 views ### A subring may have a different identity,why? [duplicate] Let$R$be a ring with unity$1_R$. Suppose$S$is a subring but it does not contain$1_R$. But still it may contain a subring unity$1_S$. For example$R=M_2(\mathbb R)$and$S$be the set of all ... 2answers 28 views ### Modular inverse: why am I allowed to use the formula :$a^{-1}\mod p \equiv a^{p-2}\mod p$[closed] Why am I allowed to do this? Do you have a reference? $$a^{-1}\mod p \equiv a^{p-2}\mod p$$ Where do I get this? From here ... 0answers 11 views ### Proof for converting one inverse trigonometric function to another I have tried to prove this result by checking the domain of two functions but every time I get entangled the last result so how to prove it 2answers 207 views ### Invertibility of elements in$A[x]$with coefficients in the Jacobson radical While solving an exercise about invertibility of elements in a polynomial ring, I came up with the following "proof" that a polynomial is invertible if its zeroth coefficient is invertible and all ... 0answers 19 views ### How to invert a multivariate series Given 2 power series of 2 variables,$y_1(x_1,x_2)=\sum\limits_{m=0}^\infty\sum\limits_{l=0}^\infty a_{m,l}x_1^mx_2^ly_2(x_1,x_2)=\sum\limits_{m=0}^\infty\sum\limits_{l=0}^\infty b_{m,l}x_1^mx_2^...
I have a matrix $M \in \mathbb{R}^{n \times n}$ whose columns are linearly independent. Hence, $M$ is invertible. How to extend this conclusion to the case where $n$ is infinite? Specifically, ...