Adjungované

Adjungované, in Czech often referred to as adjungovaná matice or adjugovaná matice, denotes the adjugate or classical adjoint of a square matrix. It is defined as the transpose of the cofactor matrix. Concretely, for an n×n matrix A, the entry in row i and column j of adj(A) is the cofactor C_{ji} of A, where C_{ji} = (-1)^{i+j} det(A_{ij}) and A_{ij} is the submatrix obtained by deleting row i and column j of A.

Key properties include:

- A times adj(A) equals adj(A) times A, and both equal det(A) times the identity matrix: A · adj(A) = adj(A) · A = det(A) I_n.

- If det(A) ≠ 0, the matrix inverse is given by A^{-1} = adj(A) / det(A). Thus the adjugate provides a way to compute the inverse via cofactors.

- For singular A (det(A) = 0), adj(A) is still defined. If rank(A) ≤ n−2, adj(A) is the zero matrix; if rank(A) = n−1, adj(A) has rank 1. In all cases, det(adj(A)) = det(A)^{n−1}.

Simple example: for A = [[a, b], [c, d]], adj(A) = [[d, -b], [-c, a]].

Computational notes: calculating adj(A) via cofactors can be expensive for large matrices. In numerical applications, direct methods for solving linear systems or computing inverses (e.g., LU decomposition) are preferred to avoid forming the full adjugate. Nonetheless, the adjugate remains a fundamental theoretical tool, linking determinants, inverses, and cofactors in linear algebra.