Let P = - 30 20 56 90 140 112 120 60 14 and A = 2 7 ω 2 - 1 - ω 1 0 - ω - ω + 1 where ω = - 1 + i 3 2 , and I 3 be the identity matrix of order 3 . If the determinant of the matrix P - 1 A P - I 3 2 is α ω 2 , then the value of α is equal to _________.

Step-by-step solution

Let M = P - 1 A P - I 2 = P - 1 A P 2 - 2 P - 1 A P + I = P - 1 A 2 P - 2 P - 1 A P + I ∵ P - 1 A P 2 = P - 1 A P P - 1 A P Multiplying with P on both sides ⇒ P M = A 2 P - 2 A P + P = A 2 - 2   A . I + I 2 P ⇒ Det P M = Det A - I 2 × P ⇒ Det P M = Det A - I 2 × Det P ⇒ Det   M = Det A - I 2 Now A - I = 1 7 w 2 - 1 - w - 1 1 0 - w - w Det A - I = w 2 + w + w + 7 - w + w 3 = - 6 w Det A - I 2 = 36 w 2 ⇒ α = 36