Adjugate matrix

In linear algebra, the adjugate, classical adjoint, or adjunct of a square matrix is the transpose of its cofactor matrix. In linear algebra, the adjugate, classical adjoint, or adjunct of a square matrix is the transpose of its cofactor matrix. The adjugate has sometimes been called the 'adjoint', but today the 'adjoint' of a matrix normally refers to its corresponding adjoint operator, which is its conjugate transpose. The adjugate of A is the transpose of the cofactor matrix C of A, In more detail, suppose R is a commutative ring and A is an n × n matrix with entries from R. The (i,j)-minor of A, denoted Mij, 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 as it is 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. The above formula implies one of the fundamental results in matrix algebra, that A is invertible if and only if det(A) is an invertible element of R. When this holds, the equation above yields The adjugate of any non-zero 1×1 matrix (complex scalar) is I = ( 1 ) {displaystyle mathbf {I} =(1)} . By convention, adj(0) = 0.