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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
Define what it means for a matrix to be diagonalizable.
💡 Hint: Think about what forms a matrix can take when diagonalized.
Question 2
Easy
What is an eigenvector?
💡 Hint: Consider how these vectors behave during transformation.
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 it mean for a matrix to be diagonalizable?
- It can be transformed into a lower triangular matrix.
- It can be represented as D = P^{-1}AP.
- It has no eigenvalues.
💡 Hint: Focus on how D changes the representation of A.
Question 2
True or False: A matrix with multiple identical eigenvalues cannot be diagonalized.
- True
- False
💡 Hint: Consider the importance of eigenvector independence.
Solve 1 more question and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
Let A = [[2, 1], [1, 2]]. Find the eigenvalues and demonstrate whether A is diagonalizable.
💡 Hint: Calculate the characteristic polynomial to find the eigenvalues.
Question 2
Consider a symmetric matrix A. Prove that it has real eigenvalues and therefore is diagonalizable.
💡 Hint: Refer to the properties of symmetric matrices and the spectral theorem.
Challenge and get performance evaluation