adjoinmatriisi

Adjoinmatriisi, also known as the adjugate matrix or classical adjoint, is a concept in linear algebra. For a square matrix A, its adjoinmatriisi, denoted as 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 calculated as (-1)^(i+j) times the determinant of the submatrix obtained by removing the i-th row and j-th column of A.

The adjoinmatriisi has important properties related to the inverse of a matrix. Specifically, the product of a matrix and its adjoinmatriisi is equal to the determinant of the matrix multiplied by the identity matrix: A * adj(A) = det(A) * I. This relationship provides a method for calculating the inverse of a matrix: A⁻¹ = (1/det(A)) * adj(A), provided that the determinant is non-zero. This formula is particularly useful for calculating the inverse of small matrices, like 2x2 or 3x3 matrices, by hand. The adjoinmatriisi also plays a role in solving systems of linear equations using Cramer's rule.