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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
Define diagonalizability in your own words.
💡 Hint: Consider the form A = PDP⁻¹.
Question 2
Easy
What is an eigenvalue?
💡 Hint: Think about how it relates to eigenvectors in the equation Av = λv.
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 is the main condition for a matrix to be diagonalizable?
- It has at least one eigenvalue.
- It has n linearly independent eigenvectors.
- It has distinct eigenvalues.
- None of the above.
💡 Hint: Think about how many vectors are needed to span the space.
Question 2
True or False: All symmetric matrices are diagonalizable.
- True
- False
💡 Hint: Recall the properties of symmetric matrices.
Solve 2 more questions and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
Given a 3x3 matrix with eigenvalues 2 (multiplicity 2) and 3 (multiplicity 1), explain whether it can be diagonalized and justify your answer.
💡 Hint: Explore geometric vs algebraic multiplicity.
Question 2
Demonstrate the process of diagonalization for a symmetric matrix with eigenvalues 4, 5, and 6. Calculate P and D, showing all steps.
💡 Hint: Use the characteristics of symmetric matrices.
Challenge and get performance evaluation