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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
Define Algebraic Multiplicity.
💡 Hint: Relate it to counting occurrences.
Question 2
Easy
What does Geometric Multiplicity represent?
💡 Hint: Think about the space spanned by eigenvectors.
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 algebraic multiplicity?
- Number of linearly independent eigenvectors
- Dimension of eigenspace
- Times eigenvalue appears in the characteristic polynomial
💡 Hint: Think about counting citations in a research paper.
Question 2
True or False: The geometric multiplicity can be greater than the algebraic multiplicity.
- True
- False
💡 Hint: Reflect on their definitions.
Solve 1 more question and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
For a matrix A with eigenvalues and their respective multiplicities: λ1=2, AM=3, GM=2; λ2=0, AM=1, GM=1. Discuss the diagonalizability of A.
💡 Hint: Refer back to the conditions that define diagonalizability and the role of eigenvalue multiplicities.
Question 2
Explain why a symmetric matrix has only real eigenvalues. How does this impact AM and GM?
💡 Hint: Consider how the properties of symmetrical matrices simplify eigenvalue calculations.
Challenge and get performance evaluation