invertibilitet

Invertibilitet (invertibility) is the property of an object under a binary operation to possess a two-sided inverse. In algebraic structures with an identity, an element a is invertible if there exists b such that ab = ba = 1. In linear algebra the term typically refers to square matrices or linear operators.

Matrices: For an n×n matrix A over a field, A is invertible if det(A) ≠ 0; equivalently rank(A) = n; equivalently there exists B with AB = BA = I; equivalently the homogeneous system Ax = 0 has only the trivial solution; equivalently the linear transformation x ↦ Ax is bijective.

Linear maps: A linear map T: V → V between finite-dimensional vector spaces of the same dimension is invertible iff it is bijective, i.e., has an inverse linear map T^{-1} with TT^{-1} = T^{-1}T = Id_V. The matrix representation of an invertible map in any basis is itself invertible.

Other contexts: In a ring, invertible elements are called units. Left- or right-invertible elements may exist without being two-sided invertible in general, though in many settings they coincide.

Computation and examples: The inverse of A can be computed by Gaussian elimination or by the adjugate divided by det. Example: A = [[1,2],[3,4]] has det -2, so A^{-1} = (-1/2) [[4,-2],[-3,1]]. A matrix with det 0, such as [[1,0],[0,0]], is not invertible. Invertibility is closely linked to solving linear systems and to the spectral property that 0 is not an eigenvalue of the matrix or operator.