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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
Define the minimal polynomial.
💡 Hint: Focus on what it means for a polynomial to be 'monic'.
Question 2
Easy
What is the characteristic polynomial's form?
💡 Hint: Recall the definition of eigenvalues.
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 of any form
- A monic polynomial of least degree such that m(A) = 0
- Only the characteristic polynomial
💡 Hint: Remember that it must satisfy m(A) = 0.
Question 2
True or False: The minimal polynomial does not necessarily divide the characteristic polynomial.
- True
- False
💡 Hint: Revisit the definitions of both polynomials.
Solve 1 more question and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
Given the matrix A = [[4, 1], [2, 3]], find its characteristic polynomial and minimal polynomial. Discuss their implications for the matrix's eigenstructure.
💡 Hint: Use the determinant and polynomial properties to derive your results.
Question 2
Consider a control system that uses the minimal polynomial m(λ) = λ^2 + 5λ + 6. Analyze the roots and discuss system stability based on these roots.
💡 Hint: Check if the roots are in the left half of the complex plane for stability.
Challenge and get performance evaluation