Let A be a 2×2 matrix A = [[1, 1], [1, -1]]. What are the eigenvalues of the matrix A¹³?

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Question

Let A be a 2×2 matrix A = [[1, 1], [1, -1]]. What are the eigenvalues of the matrix A¹³?

Accepted Answer

Correct Answer: (D) 64√2, −64√2Concept — Eigenvalues of Matrix Powers: If λ is an eigenvalue of matrix A with eigenvector x (i.e., Ax = λx), then Aᵏx = λᵏx, so λᵏ is an eigenvalue of Aᵏ.Step 1 — Find eigenvalues of A:Characteristic equation: det(A − λI) = 0det([[1−λ, 1], [1, −1−λ]]) = (1−λ)(−1−λ) − (1)(1) = −(1−λ²) − 1 = λ² − 1 − 1 = λ² − 2 = 0λ = ±√2So the eigenvalues of A are √2 and −√2.Step 2 — Eigenvalues of A¹³:Using the property above: eigenvalues of A¹³ are (√2)¹³ and (−√2)¹³.(√2)¹³ = (2^(1/2))¹³ = 2^(13/2) = 2⁶ × 2^(1/2) = 64√2(−√2)¹³ = −(√2)¹³ = −64√2 (since 13 is odd, the negative sign is preserved)Therefore, eigenvalues of A¹³ = 64√2 and −64√2.Why other options are wrong:Option A (1, −1): These would be eigenvalues of A⁰ = I, not A¹³.Option B (2√2, −2√2): (√2)³ = 2√2 — these are eigenvalues of A³, not A¹³.Option C (4√2, −4√2): (√2)⁵ = 4√2 — eigenvalues of A⁵.

Let A be a 2×2 matrix A = [[1, 1], [1, -1]]. What are the eigenvalues of the matrix A¹³?

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Let A be a 2×2 matrix A = [[1, 1], [1, -1]]. What are the eigenvalues of the matrix A¹³?

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