Find the adjoint of the given matrix and verify in each case that A. (adj A) = (adj A) =m |A|.I.
Here,
Now, we have to find adj A and for that we have to find co-factors:
a11 (co – factor of 2) = (-1)1+1(9) = (-1)2(9) = 9
a12 (co – factor of 3) = (-1)1+2(5) = (-1)3(5) = -5
a21 (co – factor of 5) = (-1)2+1(3) = (-1)3(3) = -3
a22 (co – factor of 9) = (-1)2+2(2) = (-1)4(2) = 2
Now, adj A = Transpose of co-factor Matrix
Calculating A (adj A)
= 3I
Calculating (adj A)A
= 3I
Calculating |A|.I
= (2 × 9 – 3 × 5)I
= (18 – 15)I
= 3I
Thus, A(adj A) = (adj A)A = |A|I = 3I
⇒ A(adj A) = (adj A)A = |A|I
Hence Proved
Ans.