kofaktort

Kofaktort, in mathematics, refers to the cofactor associated with a specific entry of a square matrix. The concept is central to determinant computation and to the construction of the matrix inverse. While the English term is cofactor, many languages use a cognate form such as kofaktor or kofaktort.

For an n×n matrix A, the minor M_{ij} is the determinant of the submatrix obtained by deleting row i and column j. The cofactor C_{ij} is defined as (-1)^{i+j} times the minor, i.e., C_{ij} = (-1)^{i+j} M_{ij}. The collection of cofactors forms the cofactor matrix C, and the transpose of this matrix is known as the adjugate (or adjoint) matrix adj(A). The determinant can be computed by cofactor expansion along any row or column: det(A) = sum_j a_{ij} C_{ij} for a fixed row i, or det(A) = sum_i a_{ij} C_{ij} for a fixed column j.

The inverse of A, when det(A) ≠ 0, is given by A^{-1} = (1/det(A)) adj(A). Thus cofactors play a key role in explicit inverse calculations. An example: for a 2×2 matrix A = [ [a, b], [c, d] ], the cofactors are C_{11} = d, C_{12} = -c, C_{21} = -b, C_{22} = a, and adj(A) = [ [d, -b], [-c, a] ], with det(A) = ad − bc.

Note that the term cofactor also appears in biochemistry, where it denotes a non-protein molecule required for an enzyme’s activity, a usage distinct from the mathematical cofactor described here.