Given

P be a square matrix such that P 2 = I - P,

Now solving P 2 = I - P we get,

P 4 = I + P 2 -2P

P 4 = I + I - P - 2P = 2I - 3P

Now solving, P 4 .P 2 = P 6 = 27 - 5P + 3P 2

P 6 = 2I - 5P + 3(I- P)= 57 - 8P... (i)

Similarly P 8 = 13I - 21P ... (ii)

Now adding (ii)+(i) we get,

P 8 + P 6 = 187 - 29P

Now subtracting (ii)-(i) we get,

P 8 - P 6 = 8I-13P

Now comparing with we get,

Hence, = 8 + 6 + 18-8 = 24

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