33.4 - Procedure to Diagonalize a Matrix
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Practice Questions
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What is the characteristic polynomial of the matrix A = [[1, 2], [3, 4]]?
💡 Hint: Use the formula det(A - λI) = 0.
Find the eigenvalue of the matrix A = [[5, 0], [0, 5]].
💡 Hint: This is a diagonal matrix, so the eigenvalues are on the diagonal.
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Interactive Quizzes
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What must be true for a matrix to be diagonalizable?
💡 Hint: Consider the definitions related to diagonalization criteria.
True or False: The eigenvalues must all be distinct for a matrix to be diagonalizable.
💡 Hint: Think about the conditions required for diagonalization.
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Challenge Problems
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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.
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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