Theorem 1 If A be any given square matrix of order n then A adj A adj A A A I where I is the identity matrix of order n Verification a13 a23 931 932 933 Since sum of product of elements of a row or a column with corresponding cofactors is equal to A and otherwise zero we have Let 911 912 A 9 1 922 A adj A Let 2 A11 A21 A31 then adj A A12 A22 A32 A13 A23 A33 Rationalised 2023 24 A 0 0 0 A 0 0 A 0 A 0 1 Similarly we can show adj A A A I Hence A adj A adj A A A I Definition 4 A square matrix A is said to be singular if A 0 For example the determinant of matrix A Hence A is a singular matrix Definition 5 A square matrix A is said to be non singular if A oto A Then 4 hed 2 3 4 A 5 48 3 Hence A is a nonsingular matrix We state the following theorems without proof 00 A Remark We know that adj A A A I 0 0 00 1 0 A I is zero Theorem 2 If A and B are nonsingular matrices of the same order then AB and BA are also nonsingular matrices of the same order Theorem 3 The determinant of the product of matrices is equal to product of their respective determinants that is AB A B where A and B are square matrices of the same order DETERMINANTS 89 A 0 0 Writing determinants of matrices on both sides we have O 0 A 0 A 0 0 A