For this exercise assume that the matrices are all nxn. The statement in this exercise 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. M "statement 1" is true. Justify your answer. If the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivot positions. Choose the correct answer below. O A. The statement is true. By the Invertible Matrix Theorem, if the equation Ax = 0 has a nontrivial solution, then the columns of A form a linearly independent set. Therefore, A has fewer than n pivot positions. O B. The statement is true. By the Invertible Matrix Theorem, if the equation Ax = 0 has a nontrivial solution, then matrix A is not invertible. Therefore, A has fewer than n pivot positions. OC. The statement is false. By the Invertible Matrix Theorem, if the equation Ax = 0 has a nontrivial solution, then matrix A is invertible. Therefore, A has n pivot positions. OD. The statement is false. By the Invertible Matrix Theorem, if the equation Ax =0 has a nontrivial solution, then the columns of A do not form a linearly independent set. Therefore, A has n pivot positions.

For this exercise assume that the matrices are all nxn. The statement in this exercise 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. M "statement 1" is true. Justify your answer. If the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivot positions. Choose the correct answer below. O A. The statement is true. By the Invertible Matrix Theorem, if the equation Ax = 0 has a nontrivial solution, then the columns of A form a linearly independent set. Therefore, A has fewer than n pivot positions. O B. The statement is true. By the Invertible Matrix Theorem, if the equation Ax = 0 has a nontrivial solution, then matrix A is not invertible. Therefore, A has fewer than n pivot positions. OC. The statement is false. By the Invertible Matrix Theorem, if the equation Ax = 0 has a nontrivial solution, then matrix A is invertible. Therefore, A has n pivot positions. OD. The statement is false. By the Invertible Matrix Theorem, if the equation Ax =0 has a nontrivial solution, then the columns of A do not form a linearly independent set. Therefore, A has n pivot positions.

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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