adjAij

adjAij is a notation sometimes used to denote the (i,j) entry of the adjugate (classical adjoint) of a square matrix A. The adjugate adj(A) is the transpose of the cofactor matrix. For A ∈ R^{n×n}, let M_{ij} be the minor obtained by deleting row i and column j, and define the cofactor C_{ij} = (-1)^{i+j} det(M_{ij}). Then the (i,j) entry of the adjugate is adj(A)_{ij} = C_{ji}; equivalently, adj(A) = (C_{ji})_{ij}.

Key properties include the identity A · adj(A) = adj(A) · A = det(A) I_n. If det(A) ≠ 0, adj(A) provides the inverse via adj(A) = det(A) A^{-1}. Even when det(A) = 0, adj(A) is well defined and remains useful in determinant and inverse-related formulas. The adjugate’s entries are polynomial functions of the entries of A of total degree n−1.

Computation typically proceeds by calculating all cofactors and then transposing the cofactor matrix. For a 2×2 matrix A = [[a, b], [c, d]], adj(A) = [[d, −b], [−c, a]], so adj(A)_{11} = d, adj(A)_{12} = −b, etc. In general, the cofactors determine adj(A) directly via the above relation.

Notes on notation: some sources use adjAij to denote the ij-th cofactor C_{ij} rather than the ij-th entry of adj(A); since C_{ij} = adj(A)_{ji}, care is needed to avoid confusion. Applications of the adjugate include solving linear systems via Cramer’s rule, symbolic determinant identities, and theoretical discussions of matrix inverses and rank.