Minor and Co-factors of determinants

Minor:

Let A = [a_ij] be a square matrix of order n. Then the minor M_ij of a_ij in A is the determinant of the square sub-matrix of order (n - 1) obtained by leaving i^th row and j^th column of A.

Let

M₁₁ = Minor of

M₁₂ = Minor of  

M₁₃ = Minor of  

M₂₁ = Minor of  

Similarly for the elements a₂₂, a₂₃, a₃₁, a₃₂ and a₃₃.

Cofactor:

Let A = [a_ij] be a square matrix of order n. Then the cofactor of C_ij of a_ij in A is equal to  (- 1)^i+j times the determinant of the sub-matrix of order (n - 1) obtained by leaving i^th row and j^th column of A.

C_ij = cofactor of a_ij in A.

= (- 1) ^i+j M^ij, where M_ij is minor of a_ij in A.