For A = [[2,1],[0,3]], which pair of eigenvectors is correct for the eigenvalues 2 and 3 respectively?

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Multiple Choice

For A = [[2,1],[0,3]], which pair of eigenvectors is correct for the eigenvalues 2 and 3 respectively?

Explanation:
The key idea is that eigenvectors satisfy (A − λI)v = 0 for each eigenvalue λ. For λ = 2, A − 2I = [[0, 1], [0, 1]], which gives y = 0, so eigenvectors are multiples of (1, 0). For λ = 3, A − 3I = [[−1, 1], [0, 0]], which gives −x + y = 0, so eigenvectors are multiples of (1, 1). Therefore the correct pairing is (1, 0) for the eigenvalue 2 and (1, 1) for the eigenvalue 3.

The key idea is that eigenvectors satisfy (A − λI)v = 0 for each eigenvalue λ. For λ = 2, A − 2I = [[0, 1], [0, 1]], which gives y = 0, so eigenvectors are multiples of (1, 0). For λ = 3, A − 3I = [[−1, 1], [0, 0]], which gives −x + y = 0, so eigenvectors are multiples of (1, 1). Therefore the correct pairing is (1, 0) for the eigenvalue 2 and (1, 1) for the eigenvalue 3.

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