Let A be a matrix such that A · 1 2 0 3 is a scalar matrix and 3 A = 108 . Then, A 2 equals :

Step-by-step solution

Let A = a b c d Now, A . 1 2 0 3 = a b c d 1 2 0 3 = a 2 a + 3 b c 2 c + 3 d is a scalar. ∴   c   = 0 ,   2 a + 3 b = 0 ,   a = 2 c + 3 d ⇒ a = 3 d Given, 3 A = 108 ⇒ A = 108 3 2 = 12 ⇒ a d - b c = a d = 12 ⇒ 3 d d = 12 ⇒ d 2 = 4   ∵ a = 3 d ∴   a 2 = 3 d 2 = 9 d 2 = = 9 × 4 = 36 and ∵   2 a + 3 b = 0 ⇒ b = - 2 a 3 = - 2 × 6 3 ⇒ b = - 4 Now, A 2 = a b 0 d a b 0 d = a 2 a b + b d 0 d 2 = 36 - 32 0 4