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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. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer. If A is an nxn matrix, then the equation Ax = b has at least one solution for each b in Rn. Choose the correct answer below. OA. The statement is false. By the Invertible Matrix Theorem, if Ax = b has at least one solution for each b in Rn, then the equation Ax = b has no solution. OB. The statement is true. By the Invertible Matrix Theorem, Ax = b has at least one solution for each b in Rn for all matrices of size nxn. C. The statement is false. By the Invertible Matrix Theorem, Ax = b has at least one solution for each b in R only if a matrix is invertible. D. The statement is true. By the Invertible Matrix Theorem, if Ax = b has at least one solution for each b in R, then the matrix is not invertible.

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. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer. If A is an nxn matrix, then the equation Ax = b has at least one solution for each b in Rn. Choose the correct answer below. OA. The statement is false. By the Invertible Matrix Theorem, if Ax = b has at least one solution for each b in Rn, then the equation Ax = b has no solution. OB. The statement is true. By the Invertible Matrix Theorem, Ax = b has at least one solution for each b in Rn for all matrices of size nxn. C. The statement is false. By the Invertible Matrix Theorem, Ax = b has at least one solution for each b in R only if a matrix is invertible. D. The statement is true. By the Invertible Matrix Theorem, if Ax = b has at least one solution for each b in R, then the matrix is not invertible.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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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. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer. If A is an nxn matrix, then the equation Ax = b has at least one solution for each b in Rn. Choose the correct answer below. OA. The statement is false. By the Invertible Matrix Theorem, if Ax = b has at least one solution for each b in Rn, then the equation Ax = b has no solution. OB. The statement is true. By the Invertible Matrix Theorem, Ax = b has at least one solution for each b in Rn for all matrices of size nxn. C. The statement is false. By the Invertible Matrix Theorem, Ax = b has at least one solution for each b in R only if a matrix is invertible. D. The statement is true. By the Invertible Matrix Theorem, if Ax = b has at least one solution for each b in R, then the matrix is not invertible.
Transcribed Image Text: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. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer. If A is an nxn matrix, then the equation Ax = b has at least one solution for each b in Rn. Choose the correct answer below. OA. The statement is false. By the Invertible Matrix Theorem, if Ax = b has at least one solution for each b in Rn, then the equation Ax = b has no solution. OB. The statement is true. By the Invertible Matrix Theorem, Ax = b has at least one solution for each b in Rn for all matrices of size nxn. C. The statement is false. By the Invertible Matrix Theorem, Ax = b has at least one solution for each b in R only if a matrix is invertible. D. The statement is true. By the Invertible Matrix Theorem, if Ax = b has at least one solution for each b in R, then the matrix is not invertible.
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