DETERMINANTS

Definition

Every square matrix can be associated to an expression or a number which is known as its determinant 

.If A =[a_ij] is a square matrix of order n , then the determinant of A is denoted by  det A or |A| or

Note

(1)Only square matrix have determinants.The matrices which are not square do not have determinants.

(2)The determinant of a square matrix of order 3 can be expended along any row or column .

Singular Matrix

A square matrix is a singular matrix if its determinant is zero .Otherwise , it is a non-singular matrix .

MINORS AND COFACTORS

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 .

COFACTORS

Let A =[a_ij] be a square matrix of order n .Then , the cofactor 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 .

Thus , we have

        C_ij = M_ij , if i+ j is even

                -M_ij , if i+ j is odd