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A square matrix P satisfies P^2=I-P where I is identity matrix. If P^n=5 I-8 P , then n is equal to

Options

  1. A4
  2. B7
  3. C5
  4. D6

Correct answer

D. 6

Step-by-step solution

Given P^2 = I - P . We can write P^2 = -P + I . Multiplying by P , P^3 = -P^2 + P = -(I - P) + P = -I + P + P = 2P - I . Multiplying by P again, P^4 = 2P^2 - P = 2(I - P) - P = 2I - 3P . Multiplying by P again, P^5 = 2P - 3P^2 = 2P - 3(I - P) = 2P - 3I + 3P = 5P - 3I . Multiplying by P again, P^6 = 5P^2 - 3P = 5(I - P) - 3P = 5I - 5P - 3P = 5I - 8P . Comparing P^6 = 5I - 8P with the given equation P^n = 5I - 8P , we find n = 6 . Answer: 6

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