21.12. Cayley-Hamilton Theorem

Try these first

1. 1.

   Define the Cayley-Hamilton theorem.

   Hint

   What does every square matrix do with its characteristic polynomial?

2. 2.

   What is the characteristic polynomial for a 2x2 matrix?

   Hint

   Think about the relationship between determinants and eigenvalues.

3. 3.

   What does the Cayley-Hamilton theorem state?

   • Matrices can only satisfy eigenvalues
   • Every square matrix satisfies its own characteristic equation
   • Only diagonal matrices can apply

   Hint

   Think about the fundamental properties defined by characteristic equations.

4. 4.

   True or False: The Cayley-Hamilton theorem can help simplify the computation of matrix powers.

   • True
   • False

   Hint

   Reflect on how you can express higher terms using previous terms.

5. 5.

   For a matrix C = [[5, 4], [2, 3]], derive its characteristic polynomial and verify the Cayley-Hamilton theorem.

   Hint

   Begin with calculating the determinant of C - λI first.

6. 6.

   Given a matrix D = [[0, 1], [-2, -3]], use the Cayley-Hamilton theorem to express D³ in terms of lower powers of D.

   Hint

   Think about how high powers relate to lower powers plus a constant.