We now return our discussion to matrices.

Definition. Let A = (a_ij) be a square matrix of order n and let C_ij be the cofactor of aij in the determinant |A|. Then the adjoint of A, denoted by adj A, is defined as the transpose of the cofactor matrix (C_ij)

Thus the adjoint of a square matrix A is obtained on replacing each (i,j)th element of A by the cofactor of the (j,i)th element in |A|.

Example. Find adj A, if



Solution. We have

C11 = -3    C12 =  6      C13 = -3
C21 =  5    C22 = -10   C23 =  5
C31 =  2    C32 = -4     C33 =  2

Thus,



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