In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. It is occasionally known as adjunct matrix, or "adjoint", though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.

The product of a matrix with its adjugate gives a diagonal matrix (entries not on the main diagonal are zero) whose diagonal entries are the determinant of the original matrix:

where I is the identity matrix of the same size as A. Consequently, the multiplicative inverse of an invertible matrix can be found by dividing its adjugate by its determinant.

The adjugate of A is the transpose of the cofactor matrix C of A,

In more detail, suppose R is a (unital) commutative ring and A is an n × n matrix with entries from R. The (i, j)-minor of A, denoted M_ij, is the determinant of the (n − 1) × (n − 1) matrix that results from deleting row i and column j of A. The cofactor matrix of A is the n × n matrix C whose (i, j) entry is the (i, j) cofactor of A, which is the (i, j)-minor times a sign factor:

The adjugate of A is the transpose of C, that is, the n × n matrix whose (i, j) entry is the (j,i) cofactor of A,

The adjugate is defined so that the product of A with its adjugate yields a diagonal matrix whose diagonal entries are the determinant det(A). That is,

where I is the n × n identity matrix. This is a consequence of the Laplace expansion of the determinant.