Let z = - 1 +   3 i 2 , where i =   - 1 , and r ,   s ∈ 1 ,   2 ,   3 . Let P = - z r z 2 s z 2 s z r and I be the identity matrix of order 2. Then the total number of ordered pairs ( r , s ) for which P 2 = - I is

Step-by-step solution

z = ω ( ω is complex cube root of unity) P = - ω r ω 2 s ω 2 s ω r , P 2 = - I ⇒ P 2 = ω 2 r + ω 4 s ω r + 2 s - 1 r + 1 ω r + 2 s - 1 r + 1 ω 4 s + ω 2 r =   - I = - 1 0 0 - 1 ⇒ - 1 r + 1 = 0 ⇒ r is odd ⇒   r = 1 ,   3 Also ω 2 r + ω 4 s =   - 1 ∴ r   ≠ 3 By r = 1 ⇒ ω 2 + ω 4 s = - 1 ⇒ ω 4 s = ω = ω 4 ⇒ s = 1 r ,   s = 1 ,   1 Only 1