Let P be a square matrix such that P2 = I - P. For α, β, γ, δ∈N , if Pα+Pβ=γI−29P and Pα−Pβ=δI−13P, then α+β+γ−δ is equal to

P3=P2⋅P=(I−P)P=P−(I−P)=2P−IP4=P2⋅P2=(I−P)(I−P)=2I−3PP0Similarly P5=5P−3I,P6=5I−8P,P7=13P−8I,P8=13I−21P⇒P6+P8=18I−29PP8−P6=−13P+8I,⇒α=8,β=6,γ=18,δ=8⇒α+β+γ−δ=24

Let P be a square matrix such that P2 = I - P. For α, β, γ, δ∈N , if Pα+Pβ=γI−29P and Pα−Pβ=δI−13P, then α+β+γ−δ is equal to

P3=P2⋅P=(I−P)P=P−(I−P)=2P−IP4=P2⋅P2=(I−P)(I−P)=2I−3PP0Similarly P5=5P−3I,P6=5I−8P,P7=13P−8I,P8=13I−21P⇒P6+P8=18I−29PP8−P6=−13P+8I,⇒α=8,β=6,γ=18,δ=8⇒α+β+γ−δ=24

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Let P be a square matrix such that P² = I - P. For $α, β, γ, δ \in N$ , if $P^{α} + P^{β} = γI - 29 P and P^{α} - P^{β} = δI - 13 P$ , then $α + β + γ - δ$ is equal to

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Detailed Solution

$\begin{array}{l} P^{3} = P^{2} \cdot P = (I - P) P = P - (I - P) = 2 P - I \\ P^{4} = P^{2} \cdot P^{2} = (I - P) (I - P) = 2 I - 3 P (P_{0}) \end{array}$
Similarly $P^{5} = 5 P - 3 I, P^{6} = 5 I - 8 P, P^{7} = 13 P - 8 I, P^{8} = 13 I - 21 P$
$\begin{array}{l} \Rightarrow P^{6} + P^{8} = 18 I - 29 P \\ P^{8} - P^{6} = - 13 P + 8 I, \Rightarrow α = 8, β = 6, γ = 18, δ = 8 \Rightarrow α + β + γ - δ = 24 \end{array}$