Adjungált

Adjungált refers to the adjugate or classical adjoint of a square matrix. It is a concept primarily used in linear algebra. For a given square matrix A, its adjungált is the transpose of its cofactor matrix. The cofactor matrix is formed by replacing each element of the original matrix with its corresponding cofactor. 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 adjungált matrix has several important properties and applications. One key use is in finding the inverse of a matrix. Specifically, if A is an invertible square matrix, then its inverse A⁻¹ can be calculated as (1/det(A)) * adj(A), where det(A) is the determinant of A and adj(A) is the adjungált of A. This formula highlights the close relationship between the determinant, the adjungált, and the inverse of a matrix. The adjungált also plays a role in solving systems of linear equations using Cramer's rule. Furthermore, the adjungált is defined for matrices over any commutative ring, not just fields like real or complex numbers.