Let A = a i j 2 × 2 , where a ≠ i j 0 for all i , j and A 2 = I , Let a be the sum of all diagonal elements of A and b = A Then 3 a 2 + 4 b 2 is equal to

Step-by-step solution

Let, A = p q r s A 2 = p q r s p q r s = p 2 + q r p q + q s r p + r s q r + s 2 Now given, A 2 = I ⇒ p 2 + q r p q + q s r p + r s q r + s 2 = 1 0 0 1 Now on comparing both side we get, ⇒ p 2 + q r = 1 ,   q ( p + s ) = 0 And   r ( p + s ) = 0 ,   q r + s 2 = 1 Now on solving above relation we get, q ≠ 0 ⇒ p + s = 0 ⇒ a = 0 And b = | A | = p s - q r = - p 2 - q r = - 1 ( ∵ s = - p ) ∴ 3 a 2 + 4 b 2 = 4