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21.13.1. Definition

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  1. 1.

    What is the minimal polynomial of a matrix?

    Hint

    Think about polynomials and matrices.

  2. 2.

    Does the minimal polynomial divide the characteristic polynomial?

    Hint

    Consider how these polynomials relate to eigenvalues.

  3. 3.

    What is the degree of the minimal polynomial related to?

    • It is always greater than the degree of the characteristic polynomial
    • It cannot be less than the degree of the characteristic polynomial
    • It can be equal to or less than the degree of the characteristic polynomial
    Hint

    Think about how polynomials relate in linear algebra.

  4. 4.

    Is it true that the minimal polynomial must be a factor of the characteristic polynomial?

    • True
    • False
    Hint

    Recall their mathematical relationship.

  5. 5.

    Given the matrix A = [[1, 2],[0, 1]], find its minimal polynomial and explain your reasoning.

    Hint

    Calculate the eigenvalues and their multiplicities.

  6. 6.

    If a matrix has a minimal polynomial of m(x) = (x - 3)(x - 2)^2, what can you infer about its eigenvalues and their respective multiplicities?

    Hint

    Review the factors and their powers.

Exercises

Total Questions

2

Estimated Time

4 min

Passing Score

70%

Instructions

  • Read each question carefully
  • You can use hints if you need help
  • Complete all questions before submitting

4 more questions available

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Quiz

Total Questions

2

Estimated Time

4 min

Passing Score

70%

Instructions

  • Read each question carefully
  • You can use hints if you need help
  • Complete all questions before submitting

1 more question available

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Challenge Problems

Total Questions

2

Estimated Time

4 min

Passing Score

70%

Instructions

  • Read each question carefully
  • You can use hints if you need help
  • Complete all questions before submitting