Find the eigenvalues of A = [[2, 3], [3, −6]].
Compare with the model answer
det(A − λI) = (2 − λ)(−6 − λ) − 9 = λ² + 4λ − 21 = (λ + 7)(λ − 3), so λ = 3 and λ = −7.
The Characteristic Equation
Determinants recalled, the characteristic equation det(A − λI) = 0, algebraic multiplicity, and similar matrices.
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Recalling some concepts
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Notes
λ is an eigenvalue ⇔ det(A − λI) = 0 (the characteristic equation). Multiplicity as a root = algebraic multiplicity.
Exam angle
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Notes
Trap Same eigenvalues does NOT imply similar; row-equivalent matrices usually have different eigenvalues.
Example:
