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

Question

Let z = - 1 + &#160; 3 i 2 , where i = &#160; - 1 , and r , &#160; s &#8712; 1 , &#160; 2 , &#160; 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

Aicharya · Accepted Answer

z = &#969; ( &#969; is complex cube root of unity) P = - &#969; r &#969; 2 s &#969; 2 s &#969; r , P 2 = - I &#8658; P 2 = &#969; 2 r + &#969; 4 s &#969; r + 2 s - 1 r + 1 &#969; r + 2 s - 1 r + 1 &#969; 4 s + &#969; 2 r = &#160; - I = - 1 0 0 - 1 &#8658; - 1 r + 1 = 0 &#8658; r is odd &#8658; &#160; r = 1 , &#160; 3 Also &#969; 2 r + &#969; 4 s = &#160; - 1 &#8756; r &#160; &#8800; 3 By r = 1 &#8658; &#969; 2 + &#969; 4 s = - 1 &#8658; &#969; 4 s = &#969; = &#969; 4 &#8658; s = 1 r , &#160; s = 1 , &#160; 1 Only 1 pair

Let z = - 1 +   3 i 2 , where i =   - 1 , and r ,   s ∈ 1 ,   2 ,   3 . Let P…

Solution

Let z = - 1 + &#160; 3 i 2 , where i = &#160; - 1 , and r , &#160; s &#8712; 1 , &#160; 2 , &#160; 3 . Let P…

Solution

Let z = - 1 + 3 i 2 , where i = - 1 , and r , s ∈ 1 , 2 , 3 . Let P…