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Let A be the 2×2 matrix [[1, 1], [1, 1]]. What is A raised to the 100th power?

1. [[1, 1], [1, 1]]
2. 2⁹⁹ [[1, 1], [1, 1]]
3. 2¹⁰⁰ [[1, 1], [1, 1]]
4. [[2, 2], [2, 2]]

Correct answer: 2⁹⁹ [[1, 1], [1, 1]]

Solution

Since A=[[1,1],[1,1]] gives A^2=[[2,2],[2,2]]=2A, by induction A^n = 2^(n-1) A. Hence A^100 = 2^99 [[1,1],[1,1]].

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