Inverse (mathematics)

Common types

• Additive inverse: for a number x the additive inverse is −x because x + (−x) = 0, the additive identity.
• Multiplicative inverse (reciprocal): for a nonzero number x the multiplicative inverse is 1/x, since x·(1/x) = 1, the multiplicative identity. Zero has no multiplicative inverse.
• Inverse function: a function f has an inverse f^{-1} when f is bijective; f^{-1}(y) yields the unique x with f(x)=y. Example: f(x)=2x has f^{-1}(x)=x/2. See also function inverses.
• Matrix inverse: a square matrix A is invertible if there exists B with AB = BA = I. For a 2×2 matrix [[a,b],[c,d]] the inverse exists when ad−bc ≠ 0 and equals (1/(ad−bc))[[d,−b],[−c,a]].
• Left, right and pseudoinverses: in noncommutative or noninvertible settings, an element may have a left inverse or right inverse but not both; singular matrices may admit a Moore–Penrose pseudoinverse used in least-squares problems.

Properties and existence

Where an identity element exists, an inverse (if it exists) is typically unique. In groups, every element has a unique inverse. In rings and fields, multiplicative inverses exist only for units (nonzero elements in a field). For functions, invertibility requires bijectivity; if f is not one-to-one or not onto, an inverse function does not exist globally. Composition and inversion reverse order: (g∘f)^{-1}=f^{-1}∘g^{-1} when inverses exist.

Computation and examples

Computational methods depend on the object: arithmetic inverses are simple algebraic manipulations; function inverses often require solving equations for the independent variable; matrix inversion uses determinant and adjugate for small matrices or Gaussian elimination and numerical algorithms for larger systems. In applied contexts the pseudoinverse is computed by singular value decomposition.

Applications and distinctions

Inverses appear across mathematics and applications: solving equations, undoing transformations, cryptographic algorithms that rely on modular inverses, linear system solutions using matrix inverses, and calculus where inverse functions lead to inverse derivatives. Note the term "inverse" can refer to different ideas—e.g., geometric inversion in a circle is a different notion from algebraic inverses. For further reading on related concepts see additional resources.