Adjugát

Adjugát, also spelled adjugate, is a term used in linear algebra to describe a specific matrix derived from a given square matrix. The adjugate of a matrix A, denoted as adj(A) or A*, is defined as the transpose of the cofactor matrix of A. The cofactor matrix is obtained by replacing each element of A with its corresponding cofactor, which is the determinant of the minor matrix (the matrix obtained by removing the row and column of the element) multiplied by (-1) raised to the power of the sum of the row and column indices of the element.

The adjugate matrix has several important properties. It is used in the calculation of the inverse of a matrix. Specifically, if A is an invertible matrix, then the inverse of A is given by the formula A^(-1) = adj(A) / det(A), where det(A) is the determinant of A. Additionally, the adjugate matrix satisfies the property that A * adj(A) = adj(A) * A = det(A) * I, where I is the identity matrix.

Adjugát matrices are particularly useful in solving systems of linear equations and in various applications of linear algebra. They provide a systematic way to compute the inverse of a matrix and are fundamental in the study of matrix theory.