# 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 of elliptic integral of second kind

The Wikipedia articles on elliptic integral and elliptic functions state that “elliptic functions were discovered as inverse functions of elliptic integrals.” Some elliptic functions have names and ...
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### How to prove that $A$ is invertible if and only if $A^k$ is invertible?

Let $A$ be an $n \times n$ matrix. Show that $A$ is invertible if and only if any power $A^k$ (with $k\geqslant1$) of $A$ is invertible. I've been looking over the Theorem of Invertible Matrices but ...
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### Is $A \in R^{n,n}$ invertible, if there exists an $m \in \mathbb{N}$ with $A^m = I_n$? [duplicate]

Let $R$ be a commutative ring with 1 ($1 \ne 0$). Let $A \in R^{n,n}$ be a matrix and there exists an $m \in \mathbb{N}$ with $A^m = I_n$. The number $m$ is as small as possible to hold this property. ...
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### Inverse of a Block matrix formed by two column blocks which represent orthogonal complements

I have got a matrix $B$ $m \times n$ with rank $m$. I found then the basis of its kernel and of its rowspace. Both together form a basis of $\Re^n$. With this I make the change of basis matrix $P$ ...
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1 vote
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### Sum of entries of the inverse of a positive definite matrix $I_{\ell} + A_1A_1^T + A_2A_2^T$

Let $A_1$ be an $\ell \times n$ matrix and $A_2$ be an $\ell \times m$ matrix, where $\ell \ge n, m$. The entries of $A_1$ and $A_2$ are only $0$'s and $1$'s such that Each row has exactly one $1$, ...
1 vote
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### 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 $f$?
1 vote
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### Verification of a demonstration

I need to know if the proof I made for the following problem is correct. Problem: If C is a matrix of order $3 \times 3$ such that $\text{rank}(C) = 2$, then $\text{det}(C) = 0$ Proof: If it must be ...
1 vote
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### How to calculate $\int_{0}^{2\pi}(a+\sin{x})^{-3/2}dx$ [closed]

I wonder if there is an analytic solution for the following equation: $$\int_{0}^{2\pi}(a+\sin{x})^{-3/2}dx$$ Here, $a$ is a constant. Would you please give an advice?
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### Is it true that $X(X'X)^{-1}X'-J/n$ is idempotent, where $J$ is an $n$ by $n$ matrix of ones?
$X$ is a full column rank $n$ by $p$ matrix with the first column a vector of ones. Now the I was trying to prove, from a different approach that the SSR/variance is Chi square but this means I have ...