Let (A) be a 2 × 2 matrix as given. What are the eigenvalues of the matrix…
2025
Let (A) be a 2 × 2 matrix as given.
What are the eigenvalues of the matrix 𝐴^(13) ?
- A.
(1, −1) - B.
2(2)^(1/2) and -2(2)^(1/2)
- C.
4(2)^(1/2) and -4(2)^(1/2)
- D.
64(2)^(1/2) and -64(2)^(1/2)
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Correct answer: D
Solution: Find the eigenvalues of A and then raise them to the 13th power.
Compute the characteristic polynomial: det(A − λI) = (1 − λ)(−1 − λ) − 1 = λ^2 − 2.
Thus λ^2 = 2, so the eigenvalues of A are sqrt(2) and −sqrt(2).
For any integer k, the eigenvalues of A^k are the eigenvalues of A raised to the k-th power.
(sqrt(2))^13 = 2^(13/2) = 2^6 * sqrt(2) = 64 sqrt(2).
(-sqrt(2))^13 = - (sqrt(2))^13 = -64 sqrt(2).
Therefore the eigenvalues of A^13 are 64 sqrt(2) and -64 sqrt(2).
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