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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
Define the Cayley-Hamilton theorem.
💡 Hint: What does every square matrix do with its characteristic polynomial?
Question 2
Easy
What is the characteristic polynomial for a 2x2 matrix?
💡 Hint: Think about the relationship between determinants and eigenvalues.
Practice 4 more questions and get performance evaluation
Interactive Quizzes
Engage in quick quizzes to reinforce what you've learned and check your comprehension.
Question 1
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.
Question 2
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.
Solve and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
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.
Question 2
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.
Challenge and get performance evaluation