adjugaten

The adjugate, also called the classical adjoint, of a square matrix A is the transpose of its cofactor matrix. It is denoted adj(A). In English usage, adjugate is typically used as a singular noun; the plural adjugates refers to the adjugates of several matrices.

The (i,j) entry of adj(A) is the cofactor C_{ji} of A, where C_{ij} = (-1)^{i+j} det(M_{ij}) and M_{ij} is the (n−1)×(n−1) matrix obtained by removing row i and column j from A. Thus adj(A) = C^T. For example, if A = [[a,b],[c,d]], then adj(A) = [[d, -b], [-c, a]].

A adj(A) = adj(A) A = det(A) I_n, where I_n is the identity matrix of size n. Consequently, if det(A) ≠ 0, A is invertible and A^{-1} = adj(A) / det(A). Equivalently, adj(A) = det(A) A^{-1}. The adjugate satisfies several identities, including adj(AB) = adj(B) adj(A) and adj(A^T) = adj(A)^T.

Computing the adjugate requires the cofactors of all entries, i.e., evaluating det(M_{ij}) for each i,j. This makes direct computation costly for large matrices, but it is a standard component of the inverse via the classical adjoint formula and of Cramer's rule for solving linear systems.

Beyond explicit inversion, the adjugate is involved in symbolic computation and in formulas relating determinants to inverses. It also behaves well under changes of sign and transposition, reflecting its construction from minors and cofactors.