Inverterbar

Inverterbar is a term used to describe the property of being capable of being inverted or reversed. In mathematical usage, an object is inverterbar if there exists a well-defined inverse operation that undoes its effect.

In mathematics, a function f: X → Y is inverterbar if there exists a function g: Y → X such that g(f(x)) = x for all x in X and f(g(y)) = y for all y in Y. When this condition holds, f is called invertible and g is its inverse, often denoted f⁻¹. In linear algebra, a square matrix A is inverterbar (invertible) if its determinant is nonzero; equivalently, there exists a matrix B with AB = BA = I, where I is the identity matrix. In abstract algebra, an element u of a ring is a unit if it possesses a multiplicative inverse v with uv = vu = 1.

Invertibility implies that applying the transformation is reversible and that related equations have unique solutions. Non-invertible objects fail to satisfy these conditions; for example, a non-injective function does not admit a left inverse, and a non-surjective function does not admit a right inverse.

Examples help illustrate the concept. The function f(x) = x³ on the real numbers is inverterbar, with inverse f⁻¹(y) = ∛y. The 2×2 matrix [[2, 0], [0, 3]] is inverterbar; its inverse is [[1/2, 0], [0, 1/3]]. By contrast, the matrix [[1, 0], [0, 0]] is not inverterbar because its determinant is zero.

See also: inverse function, inverse matrix, unit (algebra).