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a. If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the n - n identity matrix. b. If the columns of A span R", then the columns are linearly independent. c. If A is an nx n matrix, then the equation Ax = b has at least one solution for each b in R".

a. If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the n - n identity matrix. b. If the columns of A span R", then the columns are linearly independent. c. If A is an nx n matrix, then the equation Ax = b has at least one solution for each b in R".

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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In Exercises 11 and 12, the matrices are all n × n. Each part of the exercises is an implication of the form "If "statement 1”, then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever “statement 1" happens to be true. An implication is False if there is an instance in which "statement 2" is false but "statement 1" is true. Justify each answer. 11. a. If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the n - n identity matrix. b. If the columns of A span R", then the columns are linearly independent. c. If A is an nx n matrix, then the equation Ax = b has at least one solution for each b in R". d. If the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivot positions. e. If AT is not invertible, then A is not invertible.
Transcribed Image Text:In Exercises 11 and 12, the matrices are all n × n. Each part of the exercises is an implication of the form "If "statement 1”, then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever “statement 1" happens to be true. An implication is False if there is an instance in which "statement 2" is false but "statement 1" is true. Justify each answer. 11. a. If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the n - n identity matrix. b. If the columns of A span R", then the columns are linearly independent. c. If A is an nx n matrix, then the equation Ax = b has at least one solution for each b in R". d. If the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivot positions. e. If AT is not invertible, then A is not invertible.
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