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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
What is algebraic multiplicity?
💡 Hint: Think about how many times you can factor the polynomial.
Question 2
Easy
If a matrix has a repeated eigenvalue, how can you check if it is diagonalizable?
💡 Hint: Check the linear independence of the 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 defines the algebraic multiplicity of an eigenvalue?
- The number of linearly independent eigenvectors
- The number of times an eigenvalue appears in the characteristic polynomial
- The dimension of the eigenspace
💡 Hint: Think about the factors of the polynomial.
Question 2
True or False: A matrix with repeated eigenvalues is always diagonalizable.
- True
- False
💡 Hint: Consider an example to check this.
Solve 1 more question and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
Create a 3x3 matrix with one eigenvalue of multiplicity 3. Show its algebraic and geometric multiplicity and check if it's diagonalizable.
💡 Hint: Start with the characteristic polynomial to find AM.
Question 2
Construct a matrix that is diagonalizable with two distinct eigenvalues, and verify its multiplicities.
💡 Hint: Choose a simple diagonal matrix to illustrate diagonalizability.
Challenge and get performance evaluation