adjug

The term "adjug" is most commonly encountered in linear algebra as a shorthand for the adjugate or classical adjoint of a square matrix. The adjugate of a square matrix A, denoted adj(A), is the transpose of its cofactor matrix. The cofactor matrix is a matrix where each element is the cofactor of the corresponding element in the original matrix. The cofactor of an element a_ij is defined as (-1)^(i+j) times the determinant of the submatrix obtained by deleting the i-th row and j-th column of A.

The adjugate matrix has a significant property relating it to the inverse of a matrix. For any n x n matrix A, the product of A and its adjugate is equal to the determinant of A multiplied by the identity matrix of the same size: A * adj(A) = det(A) * I_n. This relationship is particularly useful for calculating the inverse of a matrix, especially for 2x2 and 3x3 matrices, as it provides a direct formula for the inverse: A⁻¹ = (1/det(A)) * adj(A), provided that the determinant of A is non-zero.

The concept of the adjugate is fundamental in understanding matrix properties and solving systems of linear equations using Cramer's Rule, which also relies on determinants and cofactors. While the adjugate can be computed for any square matrix, its primary practical application lies in theoretical contexts and for smaller matrices due to the computational complexity of calculating determinants of larger matrices.