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(3) Let A, B, and C be square matrices of the same order. Which of the following statements is false? (a) If A is similar to B, then A and B have the same set of eigenvalues. (b) If A is similar to B, then A" is similar to B" for all positive integers n. (c) There exist matrices A, B, and C such that A is similar to B, and B is similar to C, but A is not similar to C. (d) If A is similar to B, then det(A) = det(B). (1 mark)

(3) Let A, B, and C be square matrices of the same order. Which of the following statements is false? (a) If A is similar to B, then A and B have the same set of eigenvalues. (b) If A is similar to B, then A" is similar to B" for all positive integers n. (c) There exist matrices A, B, and C such that A is similar to B, and B is similar to C, but A is not similar to C. (d) If A is similar to B, then det(A) = det(B). (1 mark)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.5: Applications
Problem 74EQ
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(3) Let A, B, and C be square matrices of the same order. Which of the following statements is false? (a) If A is similar to B, then A and B have the same set of eigenvalues. (b) If A is similar to B, then A" is similar to B" for all positive integers n. (c) There exist matrices A, B, and C such that A is similar to B, and B is similar to C, but A is not similar to C. (d) If A is similar to B, then det(A) = det(B). (1 mark) %3D (4) Let 1 10 A = 0 2 0 0 03 and B = A' + 4A? + 31. The set of eigenvalues of B is (a) {8, 35, 120} (b) {6, 35, 40} (c) {6, 32, 20} (d) {8, 32. 80} (1 mark) (5) Let B and B' be ordered bases of a finite dimensional vector space V. Let P be the transition matrix from B to B'. Choose the correct statement: (a) P will always be a symmetric matrix. (b) P will always be a diagonal matrix. (c) P will alwways be a singular matrix (d) None of the above statements is correct. (1 mark) (6) Let A be an 3 x 3 matrix with real entries. If A is similar to the diagonal matrix D = (1 mark) then set of eigenvalucs of A is (a) (1} (b) {2} (c) {3} (d) (1.2,3}
Transcribed Image Text:(3) Let A, B, and C be square matrices of the same order. Which of the following statements is false? (a) If A is similar to B, then A and B have the same set of eigenvalues. (b) If A is similar to B, then A" is similar to B" for all positive integers n. (c) There exist matrices A, B, and C such that A is similar to B, and B is similar to C, but A is not similar to C. (d) If A is similar to B, then det(A) = det(B). (1 mark) %3D (4) Let 1 10 A = 0 2 0 0 03 and B = A' + 4A? + 31. The set of eigenvalues of B is (a) {8, 35, 120} (b) {6, 35, 40} (c) {6, 32, 20} (d) {8, 32. 80} (1 mark) (5) Let B and B' be ordered bases of a finite dimensional vector space V. Let P be the transition matrix from B to B'. Choose the correct statement: (a) P will always be a symmetric matrix. (b) P will always be a diagonal matrix. (c) P will alwways be a singular matrix (d) None of the above statements is correct. (1 mark) (6) Let A be an 3 x 3 matrix with real entries. If A is similar to the diagonal matrix D = (1 mark) then set of eigenvalucs of A is (a) (1} (b) {2} (c) {3} (d) (1.2,3}
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