ComatrixA

ComatrixA refers to the comatrix of a square matrix A, also known as the adjugate of A. The comatrixA is the transpose of the cofactor matrix. If A is an n-by-n matrix with entries aij, the cofactor Cij is (-1)^(i+j) times the determinant of the (n−1)-by-(n−1) matrix obtained by deleting row i and column j. The comatrixA is adj(A) = C^T, where C is the matrix of cofactors.

A key property is that A adj(A) = adj(A) A = det(A) I, where I is the identity matrix. Consequently, if det(A) ≠ 0, A is invertible and A^(-1) = adj(A) / det(A). The comatrixA thus provides a direct way to compute the inverse of an invertible matrix, though in practice other factorization methods (such as LU decomposition) are often preferred for numerical stability and efficiency.

Examples help illustrate the construction: for a 2-by-2 matrix A = [[a, b], [c, d]], the comatrixA is [[d, -b], [-c, a]]. For larger matrices, each entry of the comatrixA is the corresponding cofactor transposed, requiring the evaluation of many determinants of minors.

Applications of the comatrixA include theoretical results in linear algebra, explicit inverse formulas, and symbolic computations. It also appears in determinant identities and in expressions where one needs a matrix that acts as a left or right adjoint to A, scaled by det(A). In numerical practice, the comatrixA is typically used for exact symbolic calculations rather than routine numerical inversion.