Inverse

In mathematics, an inverse of a process, function, or operation is another that undoes its effect. If applying two objects in succession returns the original input, they are inverses and their composition is the identity.

Multiplicative inverse (reciprocal): for a nonzero number a, its inverse is 1/a, since a · (1/a) = 1. The additive inverse of a is -a, since a + (-a) = 0. Zero has no multiplicative inverse.

Inverse functions: a function f is invertible on a domain if it is one-to-one and onto, allowing the construction of a function f^{-1} such that f(f^{-1}(y)) = y and f^{-1}(f(x)) = x within the appropriate domains.

Matrix inverse: a square matrix A has an inverse A^{-1} when det(A) ≠ 0, with A · A^{-1} = I. The inverse is computed via methods such as Gaussian elimination or adjugation, and is used to solve linear systems and to represent inverse linear transformations.

In algebraic structures, an element g has an inverse if there exists h with g·h = h·g = e, where e is the identity. Not all structures guarantee inverses for all elements; the existence of inverses depends on the object and operation, as in groups, rings, and fields.

The term inverse is also used more broadly in sciences and logic to denote an operation or scenario that undoes a prior one, or to describe inverse problems, where one infers causes from observed effects.