or det ( ) .Let a ij be the element sitting at the intersection of the i th row and j th column of A . Deleting the i th row and j th column of A , we obtain a sub-matrix of order ( ). n − The determinant of this sub-matrix is called minor of the element a ij . It is denoted by M ij .The product of M ij and( i j is called cofactor of the element a ij .
It is denoted by A ij . Thus the cofactor of a ij is A M ij ij == −− ++ An important property connecting the elements of a square matrix and their cofactors is that the sum of the products of the entries (elements) of a row and the corresponding cofactors of the elements of the same row is equal to the determinant of the matrix; and the sum of the products of the entries (elements) of a row and the corresponding cofactors of the elements of any other row is equal to . That is, a A a A a A in jn ++ ++ ++ == == ≠≠ if if where A denotes the determinant of the square matrix A .Here A is read as “determinant of A ” and not as “ modulus of A ”. Note that A is just a real number and it can also be negative.
For instance, we have = −+ + Definition . Let A be a square matrix of order n .Then the matrix of cofactors of A is defined as the matrix obtained by replacing each element a ij of A with the corresponding cofactor A ij . The adjoint matrix of A is defined as the transpose of the matrix of cofactors of A . It is denoted byadj A .
Applications of Matrices and Determinants Note adj A is a square matrix of order