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1. Mark each statement True or False. Justify each answer, (if true, cite appropriate facts or theorems. If false, explain why or give a counterexample that shows why the statement is not true in every case). a) Any system of n linear equations in n variables has at most n solutions. b) If a system of linear equations has no free variables, then it has a unique solution. c) If an augmented matrix [A b] is transformed into [C d] by elementary row operations, then the equations Ax = b and Cx = d have exactly the same solution sets. d) If a system Ax = b has more than one solution, then so does the system Ax = 0. %3| e) The equation Ax = 0 has the trivial solution if and only if there are no free variables.

1. Mark each statement True or False. Justify each answer, (if true, cite appropriate facts or theorems. If false, explain why or give a counterexample that shows why the statement is not true in every case). a) Any system of n linear equations in n variables has at most n solutions. b) If a system of linear equations has no free variables, then it has a unique solution. c) If an augmented matrix [A b] is transformed into [C d] by elementary row operations, then the equations Ax = b and Cx = d have exactly the same solution sets. d) If a system Ax = b has more than one solution, then so does the system Ax = 0. %3| e) The equation Ax = 0 has the trivial solution if and only if there are no free variables.

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
Section6.2: Guassian Elimination And Matrix Methods
Problem 93E
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1. Mark each statement True or False. Justify each answer, (if true, cite appropriate facts or theorems. If false, explain why or give a counterexample that shows why the statement is not true in every case). a) Any system of n linear equations in n variables has at most n solutions. b) If a system of linear equations has no free variables, then it has a unique solution. c) If an augmented matrix [A b] is transformed into [C d] by elementary row operations, then the equations Ax = b and Cx = d have exactly the same solution sets. d) If a system Ax = b has more than one solution, then so does the system Ax = 0. e) The equation Ax = 0 has the trivial solution if and only if there are no free variables.
Transcribed Image Text:1. Mark each statement True or False. Justify each answer, (if true, cite appropriate facts or theorems. If false, explain why or give a counterexample that shows why the statement is not true in every case). a) Any system of n linear equations in n variables has at most n solutions. b) If a system of linear equations has no free variables, then it has a unique solution. c) If an augmented matrix [A b] is transformed into [C d] by elementary row operations, then the equations Ax = b and Cx = d have exactly the same solution sets. d) If a system Ax = b has more than one solution, then so does the system Ax = 0. e) The equation Ax = 0 has the trivial solution if and only if there are no free variables.
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