Practice Minimal Polynomial (21.13) - Linear Algebra - Mathematics (Civil Engineering -1)

Practice: Minimal Polynomial

Lesson 21.13Linear Algebra

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8 questions

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2 questions · about 1 min · pass mark 70%. Hints available.
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6 sample questions

Question 1.
Define minimal polynomial in your own words.
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Think about what it means for a polynomial to 'annihilate' a matrix.
Question 2.
What is a monic polynomial?
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What do you recall about polynomial coefficients?
3Question 3.
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.
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Focus on the purpose of the polynomial.
4Question 4.
True or False: The minimal polynomial of a matrix can never be of higher degree than the characteristic polynomial.
- True
- False
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Think about the degree relationship between the two polynomials.
Question 5.
Find the minimal polynomial for the matrix A = [[3, 1, 0], [0, 3, 1], [0, 0, 3]]. What does this tell you about A?
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Calculate the characteristic polynomial first.
Question 6.
Evaluate the implications of a minimal polynomial m(x) = x^3 for a 3x3 matrix. What structure does this imply for the eigenvalues?
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Consider the multiplicity of the eigenvalues.

4 more questions in this section. Enroll free to practise them all.