Properties of Determinants

(1)Let A =[a_ij] be a square matrix of order n , then the sum of the product of elements of any row (column ) with their cofactors is always equal to |A| or det(A) .

(2)Let A = [ a_ij] be a square matrix of order n , then the sum of the product of elements of any row (column ) with the cofactors of the corresponding elements of some other row (column) is zero .

(3)Let A =[a_ij] be a square matrix of order n , then

             |A| = |A|'

(4)If any two rows(columns) of a square matrix A of order greater than 2 are identical , then its determinant is zero , that is , |A| = 0.

(5)Let A =[a_ij] be a square matrix of order n , and let B be the matrix obtained from A by multiplying each element of a row (column ) of A by a scalar k , then

      |B| = k|A|

Note

Let A =[a_ij] be a square matrix of order n , then |kA| = kⁿ|A| , because k is common from each row of kA.

(6)Let A be a square matrix and B be a matrix obtained from A by adding to a row (column) of A a scalar multiple of another row (column ) of A , then 

      |B| = |A| .

For example -