invertibili

Invertibility is a property of an object in mathematics that allows it to be reversed by a corresponding inverse element, operation, or map. The precise meaning depends on the algebraic structure: group, ring, field, matrix, or function.

In linear algebra, a square matrix A is invertible if det(A) ≠ 0; equivalently there exists B with AB = BA = I, where I is the identity matrix. Such A defines a bijective linear transformation, and its inverse A^{-1} can be computed by the adjugate and determinant or obtained by row-reduction.

Non-invertible (singular) matrices have determinant zero and do not possess a two-sided inverse. For non-square matrices, a left inverse or right inverse may exist when the matrix has full column or row rank; in general, pseudoinverses or least-squares methods are used to approximate inverses.

In ring theory, an element a is invertible, or a unit, if there exists b with ab = ba = 1. The set of units forms a group under multiplication. Not all elements are invertible, and the invertible elements play a key role in the structure of the ring.

For functions, a function f: X → Y is invertible if it is bijective; the inverse function f^{-1}: Y → X exists and reverses f. Inverse concepts also appear in category theory and other areas, where isomorphisms generalize the notion of an invertible mapping.

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