Practice Algebraic and Geometric Multiplicities - 29.3 | 29. Eigenvalues | Mathematics (Civil Engineering -1)
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29.3 - Algebraic and Geometric Multiplicities

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Learning

Practice Questions

Test your understanding with targeted questions related to the topic.

Question 1

Easy

Define algebraic multiplicity in your own words.

💡 Hint: Think about the roots of a polynomial.

Question 2

Easy

What does geometric multiplicity represent?

💡 Hint: Consider how many independent vectors relate to an eigenvalue.

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 does algebraic multiplicity refer to?

  • The number of linearly independent eigenvectors
  • The number of times an eigenvalue appears in the characteristic polynomial
  • The dimension of the eigenspace

💡 Hint: Think about how the characteristic polynomial is defined.

Question 2

Is the geometric multiplicity ever greater than the algebraic multiplicity?

  • True
  • False

💡 Hint: Recall the inequality established in this section.

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Challenge Problems

Push your limits with challenges.

Question 1

Consider the 2x2 matrix A = | 1 2 | | 0 1 |. Determine the eigenvalues, their algebraic and geometric multiplicities, and discuss their implications for the matrix.

💡 Hint: Examine the matrix closely using the determinant for λ.

Question 2

Create a 3x3 matrix with an eigenvalue having an algebraic multiplicity of 3 but a geometric multiplicity of 1. Analyze the eigenvalue's implications on linear transformations.

💡 Hint: Focus on the structure of rows and columns leading to repeated eigenvalues.

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