Let P be a square matrix such that P 2 = I - P . For α , β , γ , δ ∈ ℕ , if P α + P β = γ l - 29 P and P α - P β = δ l - 13 P , then α + β + γ - δ is equal to

Step-by-step solution

Given, P be a square matrix such that P 2 = I - P , Now solving P 2 = I - P we get, ⇒ P 4 = I - P I - P ⇒ P 4 = I + P 2 - 2 P ⇒ P 4 = I + I - P - 2 P = 2 I - 3 P Now solving, P 4 · P 2 = P 6 = 2 I - 5 P + 3 P 2 ⇒ P 6 = 2 I - 5 P + 3 I - P = 5 I - 8 P ... i Similarly P 8 = 13 I - 21 P ... ii Now adding ii + i we get, P 8 + P 6 = 18 I - 29 P Now subtracting ii - i we get, P 8 - P 6 = 8 I - 13 P Now comparing with P α - P β = δ l - 13 P   &   P α + P &#946