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Eigenvalues solve λ² − (trace)λ + det = 0. For each one, the eigenvector satisfies (A − λI)v = 0, which for a 2 × 2 matrix can be read off as (b, λ − a) whenever b is non-zero.
Characteristic equation
λ² − (a + d)λ + (ad − bc) = 0
Eigenvector
v = (b, λ − a) when b ≠ 0
The trace is 7 and the determinant 10, so the eigenvalues are 5 and 2, with eigenvectors (1, 1) and (1, −2) up to scale.
Only up to a scalar multiple. Any non-zero multiple of an eigenvector is also an eigenvector for the same eigenvalue.