Practice Definition (21.13.1) - Linear Algebra - Mathematics (Civil Engineering -1)

Practice: Definition

Lesson 21.13.1Linear Algebra

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6 sample questions

Question 1.
What is the minimal polynomial of a matrix?
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Think about polynomials and matrices.
Question 2.
Does the minimal polynomial divide the characteristic polynomial?
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Consider how these polynomials relate to eigenvalues.
3Question 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
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Think about how polynomials relate in linear algebra.
4Question 4.
Is it true that the minimal polynomial must be a factor of the characteristic polynomial?
- True
- False
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Recall their mathematical relationship.
Question 5.
Given the matrix A = [[1, 2],[0, 1]], find its minimal polynomial and explain your reasoning.
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Calculate the eigenvalues and their multiplicities.
Question 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?
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Review the factors and their powers.

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