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Class 23 : Question 1. Class 23 : Answer 1 (A). The eigenvalues of a diagonal matrix lay along the diagonal thus the eigenvalues of D are 2 and 3. Thus the answer to the Question 23.1 is A. Class 23 : Question 2. Class 23 : Answer 2 ( C ).
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Class 23: Answer 1 (A) The eigenvalues of a diagonal matrix lay along the diagonal thus the eigenvalues of D are 2 and 3. Thus the answer to the Question 23.1 is A.
Class 23: Answer 2 ( C ) The product of a diagonal matrix with itself can easily be computed by simply multiplying the diagonal elements by themselves. By induction, this technique applies to a matrix raised to any power n. Thus the answer to the Question 23.2 is C.
Class 23: Question 3 Which of the following statements is true? • An n×n matrix with n linearly independent eigenvectors is diagonalizable. • Any diagonalizable n×n matrix has n linearly independent eigenvectors. • Both are true. • Neither is true.
Class 23: Answer 3 (C) Which of the following statements is true? • An n×n matrix with n linearly independent eigenvectors is diagonalizable. • Any diagonalizable n×n matrix has n linearly independent eigenvectors. • Both are true. • Neither is true. In order for a matrix to be diagonalized it must have n distinct eigenvectors; if it is diagonalizable then it must Have had n distinct eigenvectors. Eigenvectors are always linearly independent Thus the answer to the Question 23.3 is C.
Class 23: Question 4 Which of the following statements is true? • An n×n matrix with n distinct eigenvalues is diagonalizable. • Any diagonalizable n×n matrix has n distinct eigenvalues. • Both are true. • Neither is true.
Class 23: Answer 4 (A) Which of the following statements is true? • An n×n matrix with n distinct eigenvalues is diagonalizable. • Any diagonalizable n×n matrix has n distinct eigenvalues. • Both are true. • Neither is true. In order for a matrix to be diagonalized it must have n distinct eigenvectors; if it has n distinct eigenvalues it must have n distinct eigenvectors and thus it is diagonalizable. However, just because it is a diagonalizable matrix does not mean the matrix had n distinct eigenvalues but it must have had n distinct eigenvectors. Thus the answer to the Question 23.4 is A.
Class 23: Question 5 Which of the following statements is true? • If A is a diagonalizable matrix, then A does not have any zero eigenvalues. • If A does not have any zero eigenvalues, then A is diagonalizable. • Both are true. • Neither is true.
Class 23: Answer 5 (D) Which of the following statements is true? • If A is a diagonalizable matrix, then A does not have any zero eigenvalues. • If A does not have any zero eigenvalues, then A is diagonalizable. • Both are true. • Neither is true. Presence of a zero eigenvalue tell det(A)=0 which means A is not invertible. Whether a matrix is invertible or not is irrelevant information as to whether it is diagonalizable. Neither statement is true for ALL matrices that satisfy the given criteria. Thus the answer to the Question 23.5 is D.
Class 23: Question 6 TRUE or FALSE: All invertible matrices are diagonalizable. • TRUE. • FALSE.
Class 23: Answer 6 (2) TRUE or FALSE: All invertible matrices are diagonalizable. • TRUE. • FALSE. There are invertible matrices which are diagonalizable, and invertible matrices which are not diagonalizable. Invertibility and diagonalizability are completely unrelated. Thus the answer to the Question 23.6 is B (FALSE).
Class 23: Question 7 TRUE or FALSE: All diagonalizable matrices are invertible. • TRUE. • FALSE.
Class 23: Answer 7 (2) TRUE or FALSE: All diagonalizable matrices are invertible. • TRUE. • FALSE. There are diagonalizable matrices which are invertible, and diagonalizable matrices which are not invertible. Invertibility and diagonalizability are completely unrelated. Thus the answer to the Question 23.7 is B (FALSE).
Class 23: Answer 8 (E) It’s clear the determinants are the same: det(A)=1*2-2*3=-4=det(D) Similar matrices have the same characteristic polynomial and thus will have the same eigenvalues. Since A is similar to a diagonal matrix, it is diagonalizable, which means that AP=PD or PA=DP but NOT PA=PD. Statement 1 and 2 are true. Statement 3 is false. Thus the answer to the Question 23.8 is D.
