Adjunktenmatrix

Adjunktenmatrix, in English the adjugate (or classical adjoint) of a square matrix A, is the transpose of its cofactor matrix. For an n×n matrix with entries aij, the cofactor Cij is defined as (-1)^(i+j) times the determinant of the minor Mij obtained by removing row i and column j. The adjugate is then given by adj(A)ij = Cji, i.e., the cofactors transposed.

Key properties include that A times adj(A) equals adj(A) times A, both equal to det(A) times the identity matrix: A adj(A) = adj(A) A = det(A) I. If det(A) ≠ 0, the inverse of A can be expressed as A^(-1) = adj(A) / det(A). For a 2×2 matrix A = [[a, b], [c, d]], the adjugate is adj(A) = [[d, -b], [-c, a]].

The adjugate is closely related to solving linear systems and matrix inversion, and it plays a role in Cramer’s rule and symbolic determinant computations. It should be distinguished from the conjugate transpose (Hermitian transpose) used for complex matrices; adj(A) is not generally the same as the Hermitian transpose.

Computationally, adj(A) is obtained by computing all cofactors and transposing the resulting matrix. This can be expensive for large matrices, since it involves many determinants of minors; in numerical practice, LU or other factorization methods are often preferred for finding inverses or solving systems, unless a symbolic adjugate is specifically required.

In summary, the adjugate matrix encodes all cofactors of A, provides a direct route to A's inverse when det(A) ≠ 0, and satisfies fundamental product identities with A.