invertibile

Invertible describes the property of having an inverse in a mathematical context. Broadly, an object is invertible if there exists another object that, when combined with it under a given operation, yields an identity element. Examples include numbers, functions, and linear transformations, each with its own notion of inverse.

In arithmetic, a nonzero number a is invertible with respect to multiplication, having a multiplicative inverse a^{-1} such that a a^{-1} = a^{-1} a = 1. In a field (such as the real or complex numbers), every nonzero element is invertible under multiplication.

In linear algebra, an invertible linear transformation is a bijection, with a two-sided inverse. For a square matrix A, invertibility means there exists a matrix A^{-1} satisfying AA^{-1} = A^{-1}A = I. A matrix is invertible precisely when its determinant is nonzero. The inverse is unique, and for a 2x2 matrix, A = [[a,b],[c,d]] is invertible if ad − bc ≠ 0, with inverse (1/(ad−bc)) [[d,−b], [−c, a]].

In abstract algebra, an element u of a ring is invertible (a unit) if there exists v with uv = vu = 1. The set of all units forms a group under multiplication. Not every element is invertible; non-invertible elements include singular matrices and non-bijective maps. Some rings have zero divisors, where nonzero elements can multiply to zero, preventing invertibility.

Key properties include: the product of invertible elements is invertible, and the inverse of a product reverses the order, (AB)^{-1} = B^{-1}A^{-1}, when A and B are invertible. Invertibility is thus a central concept across many areas of mathematics.