Adjugate

In mathematics, the adjugate (or classical adjoint) of a square matrix is the transpose of the matrix of cofactors.

Given an n x n square matrix A, the adjugate of A (denoted by adj(A)) is an n x n matrix whose (i, j) entry is the (j, i) cofactor of A. In other words, adj(A) is the transpose of the matrix of cofactors of A.

The adjugate is closely related to the inverse of a matrix. In particular, if A is an invertible matrix, then adj(A) is equal to the transpose of the inverse of A multiplied by the determinant of A.

The adjugate has a number of applications in linear algebra, including the computation of determinants, the solution of linear systems, and the computation of matrix inverses.