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 R". Choose the correct answer below. A. The statement is true. By the Invertible Matrix Theorem, Ax = b has at least one solution for each b in R^ for all matrices of size nxn. B. 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. O C. 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.

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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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 R".
Choose the correct answer below.
A. The statement is true. By the Invertible Matrix Theorem, Ax = b has at least one solution for each b in R" for all matrices of size nxn.
B. 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.
O C. 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.
D. The statement is false. By the Invertible Matrix Theorem, if Ax = b has at least one solution for each b in R", then the equation Ax = b
has no solution.
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 R". Choose the correct answer below. A. The statement is true. By the Invertible Matrix Theorem, Ax = b has at least one solution for each b in R" for all matrices of size nxn. B. 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. O C. 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. D. The statement is false. By the Invertible Matrix Theorem, if Ax = b has at least one solution for each b in R", then the equation Ax = b has no solution.
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