Let P be a square matrix such that . For , if and , then is equal to:

24 Explanation: \begin{aligned} & \mathrm{P}^{2}=\mathrm{I-P} & ...(1) \\ & \mathrm{P}^{\alpha}+\mathrm{P}^{\beta}=\gamma\mathrm{I}-29\mathrm{P} & ...(2) \\ & \mathrm{P}^{\alpha}-\mathrm{P}^{\beta}=\mathrm{\delta I}-13\mathrm{P} & ...(3) \\ & \mathrm{By~(1)} \\ & \mathbf{P}^{2}=\mathbf{I}-\mathbf{P} \\ & \Rightarrow\mathrm{P}^{4}=\left(\mathrm{I-P}\right)^{2} \\ & \Rightarrow\mathrm{P}^{4}=\mathrm{I}+\mathrm{P}^{2}-2\mathrm{P} & [\because\mathrm{P}^{2}=\mathrm{I-P}] \\ & \Rightarrow\mathrm{P}^{8}=\left(2\mathrm{I-3P}\right)^{2} \\ & \Rightarrow\mathrm{P}^{8}=4\mathrm{I}^{2}+9\mathrm{P}^{2}-12\mathrm{P} \\ & \Rightarrow\mathrm{P}^{8}=13\mathrm{I}-21\mathrm{P} & ...(1) \\ & \mathrm{P^{6}=P^{4}.P^{2}=(2I-3P)~(I-P)} \\ & \mathrm{P}^{6}=5\mathrm{I}-8\mathrm{P} & ...(2) \\ & (1)+(2)\mathrm{~and~}(1)-(2) \\ & \m

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JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let P be a square matrix such that $P^{2} = I - P$ . For $α, β, γ, δ \in N$ , if $F^{α} + P^{β} = λ I - 29 P$ and $F^{α} - P^{β} = δ I - 13 P$ , then $α + β + γ - δ$ is equal to:

JEE 2023 — Mathematics PYQ

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