Practice Example (33.5) - Diagonalization - Mathematics (Civil Engineering -1)

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6 sample questions

Question 1.
What is the characteristic polynomial of the matrix A = [[4, 1], [2, 3]]?
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Use det(A - λI) to derive it.
Question 2.
Define what eigenvalues are in relation to a matrix.
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Think about the effect of a transformation on vectors.
3Question 3.
What is the significance of the characteristic polynomial in diagonalization?
- It provides eigenvalues
- It gives eigenvectors
- It's irrelevant
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Think about where eigenvalues come from.
4Question 4.
True or False: Every matrix can be diagonalized.
- True
- False
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Consider the conditions for diagonalizability.
Question 5.
Given the matrix A = [[5, 1], [2, 4]], find its eigenvalues and demonstrate the diagonalization process.
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Follow the characteristic polynomial and eigenvector process.
Question 6.
Consider the matrix A = [[3, 2], [4, 1]]. Is it diagonalizable? Justify your answer with calculations.
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Always compare the algebraic and geometric multiplicities.

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