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A 3x3 square matrix M satisfies M² = I - M, where I is the 3x3 identity matrix. If Mⁿ = 2I - 3M for some positive integer n, then n equals:
- 4
- 5
- 6
- 7
Correct answer: 4
Solution
From M² = I - M, any power Mⁿ = aₙ * I + bₙ * M. Recurrence: aₙ₊₁ = bₙ, bₙ₊₁ = aₙ - bₙ. Starting from M¹: (a,b)=(0,1). M²:(1,-1). M³:(-1,2). M⁴:(2,-3). So M⁴ = 2I - 3M, giving n=4.
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