adjugation

Adjugation refers to the construction of the adjugate (or classical adjoint) of a square matrix. The adjugate of A, denoted adj(A), is the transpose of the cofactor matrix. The cofactor matrix is formed by the cofactors C_ij = (-1)^{i+j} det(M_ij), where M_ij is obtained by deleting row i and column j from A.

A central identity is A adj(A) = adj(A) A = det(A) I. Consequently, if det(A) ≠ 0, the inverse is given by A^{-1} = adj(A) / det(A). This relationship provides a direct way to compute inverses using determinants of minors.

For computation, the adjugate entries are derived from the cofactors. In the simple case A = [a b; c d], adj(A) = [d -b; -c a]. In general, adj(A) is a polynomial in the entries of A of degree n−1, since each cofactor is a determinant of an (n−1)×(n−1) submatrix.

Rank properties and edge cases: If det(A) = 0, adj(A) may vanish. Specifically, if rank(A) ≤ n−2, then adj(A) = 0. If rank(A) = n−1, adj(A) ≠ 0 and has rank 1. If rank(A) = n (i.e., A is invertible), adj(A) ≠ 0 and equals det(A) A^{-1}.

Applications include solving linear systems via Cramer’s rule and inverse formulas, as well as various determinant identities. The adjugate also appears in broader contexts within linear and multilinear algebra and in the study of polynomial functions of matrix entries.