Practice Procedure To Diagonalize A Matrix (33.4) - Diagonalization - Mathematics (Civil Engineering -1)

Practice: Procedure to Diagonalize a Matrix

Lesson 33.4Diagonalization

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6 sample questions

Question 1.
What is the characteristic polynomial of the matrix A = [[1, 2], [3, 4]]?
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Use the formula det(A - λI) = 0.
Question 2.
Find the eigenvalue of the matrix A = [[5, 0], [0, 5]].
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This is a diagonal matrix, so the eigenvalues are on the diagonal.
3Question 3.
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.
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Consider the definitions related to diagonalization criteria.
4Question 4.
True or False: The eigenvalues must all be distinct for a matrix to be diagonalizable.
- True
- False
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Think about the conditions required for diagonalization.
Question 5.
Given the matrix A = [[1, 2], [2, 1]], determine whether it is diagonalizable. Calculate its eigenvalues and eigenvectors step by step.
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Ensure you correctly compute the determinant and solve the eigenvalue equation.
Question 6.
Consider the matrix B = [[1, 1], [0, 1]]. Prove that it cannot be diagonalized, discussing the implications of repeated eigenvalues.
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Reflect on the relationship between eigenvalues and the linear independence of eigenvectors.

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