Invertibil

Invertibil, or invertibility, is a concept in mathematics describing when an operation or object can be reversed uniquely. In a linear algebra context, it most often refers to square matrices, linear transformations, or functions that have an inverse operation.

For a square matrix A, invertibility means there exists a matrix B such that AB = BA = I, where I is the identity matrix. When this occurs, A is said to be invertible and its inverse is denoted A^−1. Several equivalent conditions characterize invertibility: the determinant det(A) is nonzero, the rank of A is full (equal to the matrix size), and the columns (or rows) are linearly independent. These equivalences imply that the linear system Ax = b has a unique solution for every vector b. The inverse can be computed in several ways, including Gaussian elimination, the adjugate formula A^−1 = (1/det A) adj(A) for nonsingular A, or decompositions such as LU or QR.

In linear transformations, a map T: V → V is invertible if and only if it is bijective; then there exists a inverse map T^−1 that is also linear. In practical terms, an invertible transformation preserves information and can be reversed.

More generally, a function is invertible if it is one-to-one on its domain and onto its image, guaranteeing a well-defined inverse function f^−1. In calculus, the inverse function theorem gives conditions under which a locally invertible function exists, typically where the derivative does not vanish.

Objects that lack invertibility are called singular or non-invertible; for matrices, this means det(A) = 0. Invertibility is fundamental to solving systems, transforming coordinates, and understanding reversible processes.