If A = 3 - 1 4 5 , then find a d j ( a d j A ) .

We have to find a d j ( a d j A ) . Given that A = 3 - 1 4 5 We know that for a non-singular matrix B of order n , we have B a d j B = B I . . . . . . . . . . . . . ( 1 ) replacing B by a d j A in eqn. ( 1 ) ; we have a d j A a d j ( a d j A ) = adj A I Pre-multiplying by A both sides, ( A a d j A ) a d j ( a d j A ) = A adj A [ ∵ A · I = A ] ⇒ A I a d j ( a d j A ) = A adj A [ ∵ ( Aadj A ) = A I ] ⇒ a d j ( a d j A ) = A adj A A ⇒ a d j ( a d j A ) = A A n - 1 A [ ∵ A ≠ 0 , adj A = A n - 1 ] ⇒ a d j ( a d j A ) = A n - 2 A . . . . . . . . . . . . . . . . . ( 2 ) And A = 3 - 1 4 5 = 15 + 4 = 19 ≠ 0 From ( 2 ) , we have ⇒ a d j ( a d j A ) = ( 19 ) 2 - 2 3 - 1 4 5 = 3 - 1 4 5 .

If A is a non-singular matrix, prove that A - 1 = 1 A .

Since it is given A is a non-singular matrix, therefore A ≠ 0 And according to property of inverse of matrix, AA - 1 = I = A - 1 A Taking determinant both sides, ⇒ AA - 1 = I ⇒ A A - 1 = 1 [ ∵ AB = A B and I = 1 ] ⇒ A - 1 = 1 A Hence, it is proved that If A is a non-singular matrix, then A - 1 = 1 A .

If A = 3 - 5 4 2 , find A ( a d j A ) .

We have to find the value of A ( a d j A ) . Given matrix A = 3 - 5 4 2 Let C ij be cofactor of a ij in A . Then, the cofactors of elements of A are given by C 11 = 2 , C 12 = - ( 4 ) = - 4 , C 21 = - ( - 5 ) = 5 , C 22 = 3 ∴ adjA = C 11 C 12 C 21 C 22 T = 2 - 4 5 3 T = 2 5 - 4 3 Now, A ( adj A ) = 3 - 5 4 2 2 5 - 4 3 = 6 + 20 15 - 15 8 - 8 20 + 6 = 26 0 0 26 This is the required value of A ( a d j A ) .