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Eigenvalues of a 2 × 2 matrix

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Useful for: IB Math AI HL

Learn it λ is an eigenvalue of A if det(A − λI) = 0. For A = [a b; c d] this is λ² − (a + d)λ + (ad − bc) = 0. Eigenvalues describe long-run behaviour, like the steady state of a Markov chain.

Example: Eigenvalues of [4 1; 2 3]?

  1. λ² − 7λ + 10 = 0 → (λ − 5)(λ − 2) = 0.
  2. λ = 5 and λ = 2.

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