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If A = [[1, 2], [2, -1]], then which ONE of the following is A⁸ ?

1. [[25, 0], [0, 25]]
2. [[125, 0], [0, 125]]
3. [[625, 0], [0, 625]]
4. [[3125, 0], [0, 3125]]

Correct answer: [[625, 0], [0, 625]]

Solution

For A=[[1,2],[2,-1]], A^2 = [[5,0],[0,5]] = 5I. Therefore A^8 = (A^2)^4 = 5^4 I = 625 I = [[625,0],[0,625]], not 25I.

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