A square matrix $P$ satisfies $P^2 = I - P$, where $I$ is the identity matrix. If $P^n = 5I - 8P (n \in N)$, then minimum value of $n$ is equal to
Step-by-Step Solution
Key Concept: Use the recurrence relation P² = I - P to find a pattern for successive powers of P, then match the form P^n = 5I - 8P by computing powers systematically until the pattern emerges.
<p><strong>Step 1:</strong> Start with the given relation P² = I - P, which can be rewritten as P² + P - I = 0.</p><p><strong>Step 2:</strong> Calculate successive powers of P using this recurrence relation.</p><p>P² = I - P</p><p>P³ = P · P² = P(I - P) = P - P² = P - (I - P) = 2P - I</p><p>P⁴ = P · P³ = P(2P - I) = 2P² - P = 2(I - P) - P = 2I - 3P</p><p>P⁵ = P · P⁴ = P(2I - 3P) = 2P - 3P² = 2P - 3(I - P) = 5P - 3I = -3I + 5P</p><p>P⁶ = P · P⁵ = P(5P - 3I) = 5P² - 3P = 5(I - P) - 3P = 5I - 8P</p><p><strong>Step 3:</strong> Identify the pattern. We need P^n = 5I - 8P. From Step 2, we found that P⁶ = 5I - 8P.</p><p><strong>Step 4:</strong> Verify this is the minimum value of n. Checking n = 1, 2, 3, 4, 5 from our calculations above, none match the form 5I - 8P. Only P⁶ satisfies this condition.</p><p><strong>∴ Answer: C</strong></p>
Correct Answer: C