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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
What is algebraic multiplicity?
💡 Hint: Consider how many times you can factor the polynomial.
Question 2
Easy
Define geometric multiplicity.
💡 Hint: Think about how many directions correspond to that eigenvalue.
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 algebraic multiplicity refer to?
- The number of eigenvectors for an eigenvalue
- The number of times an eigenvalue is a root of the characteristic polynomial
- The stability of the matrix
💡 Hint: Focus on root counting!
Question 2
True or False: Geometric multiplicity can be greater than algebraic multiplicity.
- True
- False
💡 Hint: Remember the inequalities we've discussed.
Solve 1 more question and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
Consider a matrix with characteristic polynomial p(λ) = (λ - 3)³(λ - 2) = 0. Determine the algebraic and geometric multiplicities for each eigenvalue.
💡 Hint: Don't forget the concepts and check eigenspace dimensions!
Question 2
If an eigenvalue's algebraic multiplicity is 5 and geometric is 3, what can you infer about the matrix properties in terms of stability?
💡 Hint: Link it back to applications in structural assessments.
Challenge and get performance evaluation