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Given the matrix M = [[5/2, 3/2], [−3/2, −1/2]], which of the following represents the value of M raised to the power 2022?

1. [[3034, 3033], [−3033, −3032]]
2. [[3034 − 3033, 3033 − 3032], [3033 − 3032, −3032 − 3031]]
3. [[3033, 3032], [−3032, −3031]]
4. [[3032, 3031], [−3031, −3030]]

Correct answer: [[3034, 3033], [−3033, −3032]]

Solution

The matrix M has eigenvalues that allow it to be expressed in a diagonalizable form. Repeated multiplication of M by itself (raising it to a power) results in a pattern where the entries grow linearly with the power. This leads to the given result for M raised to the power 2022.

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