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14. Which statements are true, and which are false? If the statement is false, provide a specific counterexample, and show why it works. If the statement is true, explain why. (a) AB = BA for any n x n matrices A, B. (b) Let A, B be nx n matrices. If AB = BA, then (A + B)(A − B) = A² - B². (c) Let A, B be 2 x 2 matrices. If A² = AB and det(A) = 4, then A = B. (d) Let A, B be two n x n matrices. If AB = 0, then A=0 or B = 0.

14. Which statements are true, and which are false? If the statement is false, provide a specific counterexample, and show why it works. If the statement is true, explain why. (a) AB = BA for any n x n matrices A, B. (b) Let A, B be nx n matrices. If AB = BA, then (A + B)(A − B) = A² - B². (c) Let A, B be 2 x 2 matrices. If A² = AB and det(A) = 4, then A = B. (d) Let A, B be two n x n matrices. If AB = 0, then A=0 or B = 0.

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.5: Determinants
Problem 86E
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14. Which statements are true, and which are false? If the statement is false, provide a specific counterexample, and show why it works. If the statement is true, explain why. (a) AB = BA for any n x n matrices A, B. (b) Let A, B be n x n matrices. If AB = BA, then (A + B)(A − B) = A² - B². (c) Let A, B be 2 x 2 matrices. If A² = AB and det(A) = 4, then A = B. (d) Let A, B be two n x n matrices. If AB = 0, then A = 0 or B = 0.
Transcribed Image Text:14. Which statements are true, and which are false? If the statement is false, provide a specific counterexample, and show why it works. If the statement is true, explain why. (a) AB = BA for any n x n matrices A, B. (b) Let A, B be n x n matrices. If AB = BA, then (A + B)(A − B) = A² - B². (c) Let A, B be 2 x 2 matrices. If A² = AB and det(A) = 4, then A = B. (d) Let A, B be two n x n matrices. If AB = 0, then A = 0 or B = 0.
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