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3 cards from this section. Quick, good the night before a test.
Define the Cayley-Hamilton theorem.
What does every square matrix do with its characteristic polynomial?
What is the characteristic polynomial for a 2x2 matrix?
Think about the relationship between determinants and eigenvalues.
What does the Cayley-Hamilton theorem state?
Think about the fundamental properties defined by characteristic equations.
True or False: The Cayley-Hamilton theorem can help simplify the computation of matrix powers.
Reflect on how you can express higher terms using previous terms.
For a matrix C = [[5, 4], [2, 3]], derive its characteristic polynomial and verify the Cayley-Hamilton theorem.
Begin with calculating the determinant of C - λI first.
Given a matrix D = [[0, 1], [-2, -3]], use the Cayley-Hamilton theorem to express D³ in terms of lower powers of D.
Think about how high powers relate to lower powers plus a constant.
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