invertibility

Invertibility is the property of an object that possesses an inverse operation under a given binary operation. In mathematics, this idea appears in several contexts, notably with functions, matrices and other algebraic structures.

In the context of functions, a function f from a set X to a set Y is invertible precisely when it is a bijection. In that case there exists a unique inverse function f^{-1} that undoes f. If f is not bijective, it may still have a left inverse (an introduction of an inverse on the image of f that exists when f is injective) or a right inverse (an inverse on the codomain that exists when f is surjective); a two-sided inverse exists exactly when f is bijective.

In linear algebra, a square matrix A is invertible if there exists a matrix B such that AB = BA = I, where I is the identity matrix. This occurs if and only if det(A) ≠ 0, which is also equivalent to having full rank (rank n for an n-by-n matrix). The inverse is denoted A^{-1}, and it can be computed by formulas such as A^{-1} = (1/det A) adj(A) in the 2×2 and classical cases, or via more general algorithms like Gaussian elimination or LU decomposition. A non-invertible matrix is called singular.

Beyond square matrices, concepts such as left and right inverses arise for non-square matrices: a matrix with full column rank admits a left inverse, while one with full row rank admits a right inverse. The Moore–Penrose pseudoinverse provides a least-squares generalization of inversion for arbitrary matrices.

In abstract algebra, an element with a multiplicative inverse is called a unit. In a group every element has an inverse, while in a ring not all elements are invertible. Invertibility underpins solving linear systems, change of basis, and many structural results across mathematics.