Practice Procedure to Diagonalize a Matrix - 33.4 | 33. Diagonalization | Mathematics (Civil Engineering -1)
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33.4 - Procedure to Diagonalize a Matrix

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Learning

Practice Questions

Test your understanding with targeted questions related to the topic.

Question 1

Easy

What is the characteristic polynomial of the matrix A = [[1, 2], [3, 4]]?

💡 Hint: Use the formula det(A - λI) = 0.

Question 2

Easy

Find the eigenvalue of the matrix A = [[5, 0], [0, 5]].

💡 Hint: This is a diagonal matrix, so the eigenvalues are on the diagonal.

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 must be true for a matrix to be diagonalizable?

  • It has n linearly independent eigenvectors.
  • It must быть symmetric.
  • It contains only positive eigenvalues.

💡 Hint: Consider the definitions related to diagonalization criteria.

Question 2

True or False: The eigenvalues must all be distinct for a matrix to be diagonalizable.

  • True
  • False

💡 Hint: Think about the conditions required for diagonalization.

Solve and get performance evaluation

Challenge Problems

Push your limits with challenges.

Question 1

Given the matrix A = [[1, 2], [2, 1]], determine whether it is diagonalizable. Calculate its eigenvalues and eigenvectors step by step.

💡 Hint: Ensure you correctly compute the determinant and solve the eigenvalue equation.

Question 2

Consider the matrix B = [[1, 1], [0, 1]]. Prove that it cannot be diagonalized, discussing the implications of repeated eigenvalues.

💡 Hint: Reflect on the relationship between eigenvalues and the linear independence of eigenvectors.

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