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A square matrix P obeys the relation P² = I - P, where I denotes the identity matrix. If Pⁿ = 5I - 8P, then the value of n is

1. 4
2. 5
3. 6
4. 7

Correct answer: 6

Solution

From P^2 = I - P, each power has the form P^k = a_k I + b_k P with (a,b): P^1=(0,1), P^2=(1,-1), P^3=(-1,2), P^4=(2,-3), P^5=(-3,5), P^6=(5,-8). Thus P^6 = 5I - 8P, so n = 6.

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