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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
Define minimal polynomial in your own words.
💡 Hint: Think about what it means for a polynomial to 'annihilate' a matrix.
Question 2
Easy
What is a monic polynomial?
💡 Hint: What do you recall about polynomial coefficients?
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 minimal polynomial of a matrix?
- A polynomial that always has distinct roots.
- The polynomial of least degree that annihilates the matrix.
- A polynomial that has a non-zero leading coefficient.
💡 Hint: Focus on the purpose of the polynomial.
Question 2
True or False: The minimal polynomial of a matrix can never be of higher degree than the characteristic polynomial.
- True
- False
💡 Hint: Think about the degree relationship between the two polynomials.
Solve and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
Find the minimal polynomial for the matrix A = [[3, 1, 0], [0, 3, 1], [0, 0, 3]]. What does this tell you about A?
💡 Hint: Calculate the characteristic polynomial first.
Question 2
Evaluate the implications of a minimal polynomial m(x) = x^3 for a 3x3 matrix. What structure does this imply for the eigenvalues?
💡 Hint: Consider the multiplicity of the eigenvalues.
Challenge and get performance evaluation