# 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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### Prove it or give a counterexample to the statement [closed]

Prove it or give a counterexample to the statement If $A$ is an invertible $n\times n$ matrix, then the number of solutions of $Ax = b$ depends on the vector $b \in\mathbb R^n$.
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### Invertibility of Mixing Matrix $M$ in $A=CMR$

I'm interested in solving the following problem from Strang's Linear Algebra and Learning from Data: If $C$ and $R$ contain bases for the column space and row space of $A$, why does $A=CMR$ for ...
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### Inverting a sum of Kronecker's delta

I am trying to evaluate the following expression: \begin{equation} \begin{split} A=\sum_{\ell_1\ell_2\ell_3}\sum_{\ell_1'\ell_2'\ell_3'}C_{\ell_1\ell_2\ell_3}\Big[&\delta_{\ell_1\ell_1'}\delta_{\...
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### Schur complement like operation on a singular matrix

For the classical definition of matrix inversion by Schur complement, given by: \begin{aligned} M^{-1}=\left[\begin{array}{ll} A & B \\ C & D \end{array}\right]^{-1} &=\left(\left[\begin{...
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### Inverse of a Fractional Exponent?

Hello Mathematics Stack Exchange, I'm currently a Grade $11$ Math Student and to train for this year's exam I am going through a worked video of the previous years' exams to get a better understanding ...
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### Gradient of the product of an inverse matrix and a vector

I would like to expand the gradient of the following product of the matrix $M$ and the vector $v$ using the product rule $$\nabla (M^{-1} \cdot v) = M^{-1} \cdot \nabla v + \nabla (M^{-1}) \cdot v$$ ...
### Prove that $X \subset Y \implies f^{-1}(X) \subset f^{-1}(Y)$ .... Is invertibility assumed?
Given a function f: A→B where X, Y $\subset$ B, prove that $X \subset Y \implies f^{-1}(X) \subset f^{-1}(Y)$ I was able to go ahead and prove this statement by applying the definition of the pre-...
### Finding $\mathrm{\int_{\Bbb R} a^{cosh(x)} dx}$
After the success and great answer by @metamorphy of my $$\mathrm{\int_{-\pi}^0 a^{csc(x)}dx}$$ question, I experimented a bit more and found this nice graph. This looks almost like a gamma function ...