Invertible

Invertible describes an object that has an inverse under a defined operation; in many mathematical contexts, invertibility means there exists a two-sided inverse that cancels the original operation, so that composing the two together yields an identity transformation or element.

Functions: A function f: A → B is invertible if it is bijective, meaning it is one-to-one and onto. In that case there exists a unique inverse function f^{-1}: B → A, with f^{-1}(f(x)) = x and f(f^{-1}(y)) = y for all applicable x and y. Non-bijective functions do not have inverses, though restricting the domain can sometimes render a non-injective function invertible on that restricted domain.

Matrices and linear maps: A square matrix A is invertible if there exists a matrix B such that AB = BA = I, where I is the identity matrix. Equivalently, det(A) ≠ 0 and A has full rank. The inverse is denoted A^{-1}. The corresponding linear transformation is invertible precisely when it is bijective, meaning it has a trivial kernel and its image is the whole target space.

Algebraic structures: In a ring, an element is invertible if there exists another element that multiplies with it to give the multiplicative identity; such elements are called units. The set of units forms a group under multiplication. In fields, every nonzero element is invertible. In broader contexts, invertibility describes the possibility of reversing a process or operation.