factorcofactor

Factorcofactor is a term used in mathematics to describe a combined construct that arises from considering both a matrix factorization and the matrix’s cofactors. In this sense, a factorcofactor of a square matrix A refers to an ordered pair or a small collection of related objects that capture both a factorization of A and its cofactor information, with the intent of streamlining certain computational tasks such as determinant, inverse, or adjugate calculations.

A practical formulation treats the factorcofactor of A as a pair (F, C), where F denotes a valid factorization of A (for example A = L U or A = P A Q), and C denotes the cofactor matrix cof(A) of A. The exact interpretation can be context dependent: F might be chosen to optimize a particular computation, while C provides the cofactors needed for determinant and adjugate-related operations. In settings where multiple factorizations are possible, different factorcofactor representations may be used for the same matrix.

Key properties follow from the underlying facts that A = F G for some factorization, and that det(A) and adj(A) are linked to A's inverse when it exists. If A = F G, then det(A) = det(F) det(G). The cofactor matrix satisfies adj(A) = det(A) A^{-1} when A is invertible, and the factorcofactor framework can be employed to derive det(A) and inverse-related quantities in tandem.

Applications of the factorcofactor concept are mainly pedagogical or algorithmic, providing a unified viewpoint for algorithms that require both factorization and cofactors. The term is not standard in literature; precise usage depends on a clear, explicit definition.

See also: Cofactor, Adjugate, Matrix factorization, Determinant, Inverse.