Let P be a square matrix such that P ^ 2 = I - P . For α, β, γ, ∈ ℕ...

Given,

$P$ be a square matrix such that ${P}^{2} = I - P$,

Now solving ${P}^{2} = I - P$ we get,

$⇒ {P}^{4} = \left(I - P\right)\left(I - P\right)$

$⇒ {P}^{4} = I + {P}^{2} - 2P$

$⇒ {P}^{4} = I + I - P - 2P = 2I - 3P$

Now solving, ${P}^{4} \cdot {P}^{2} = {P}^{6} = 2I - 5P + 3{P}^{2}$

$⇒ {P}^{6} = 2I - 5P + 3\left(I - P\right) = 5I - 8P$... \left(i\right)$</p><p>Similarly${P}^{8} = 13I - 21P$... \left(ii\right)$

Now adding \left(ii\right) + \left(i\right) we get,

${P}^{8} + {P}^{6} = 18I - 29P$

Now subtracting \left(ii\right) - \left(i\right) we get,

${P}^{8} - {P}^{6} = 8I - 13P$

Now comparing with ${P}^{\alpha} - {P}^{\beta} = \deltal - 13P$\text{and } {P}^{\alpha} + {P}^{\beta} = \gammal - 29P we get,

$\alpha = 8, \beta = 6, \gamma = 18, \delta = 8$

Hence, $\alpha + \beta + \gamma - \delta = 8 + 6 + 18 - 8 = 24$