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 the equation Ax=b has at least one solution for each b in R", then the solution is unique for each b. Choose the correct answer below. O A. The statement is true. By the Invertible Matrix Theorem, if Ax=b has at least one solution for each b in R", then matrix A is invertible. If A is invertible, then according to the invertible matrix theorem the solution is unique for each b. OB. The statement is true, but only for x +0. By the Invertible Matrix Theorem, if Ax=b has at least one solution for each b in R", then the equation Ax=0 does not only have the trivial solution. OC. The statement is false. By the Invertible Matrix Theorem, if Ax=b has at least one solution for each b in R", then the linear transformation x → Ax does not map R onto R. O D. The statement is false. By the Invertible Matrix Theorem, if Ax=b has at least one solution for each b in R", then matrix A is not invertible. If A is not invertible, then according to the Invertible Matrix Theorem, the solution is not unique for each b.
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 the equation Ax=b has at least one solution for each b in R", then the solution is unique for each b. Choose the correct answer below. O A. The statement is true. By the Invertible Matrix Theorem, if Ax=b has at least one solution for each b in R", then matrix A is invertible. If A is invertible, then according to the invertible matrix theorem the solution is unique for each b. OB. The statement is true, but only for x +0. By the Invertible Matrix Theorem, if Ax=b has at least one solution for each b in R", then the equation Ax=0 does not only have the trivial solution. OC. The statement is false. By the Invertible Matrix Theorem, if Ax=b has at least one solution for each b in R", then the linear transformation x → Ax does not map R onto R. O D. The statement is false. By the Invertible Matrix Theorem, if Ax=b has at least one solution for each b in R", then matrix A is not invertible. If A is not invertible, then according to the Invertible Matrix Theorem, the solution is not unique for each b.
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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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 the equation Ax = b has at least one solution for each b in R, then the solution is unique for each b.
Choose the correct answer below.
A. The statement is true. By the Invertible Matrix Theorem, if Ax=b has at least one solution for each b in R", then matrix A is
invertible. If A is invertible, then according to the invertible matrix theorem the solution is unique for each b.
OB. The statement is true, but only for x 0. By the Invertible Matrix Theorem, if Ax=b has at least one solution for each b in R, then
the equation Ax = 0 does not only have the trivial solution.
OC. The statement is false. By the Invertible Matrix Theorem, if Ax=b has at least one solution for each b in R, then the linear
transformation x → Ax does not map R onto R".
OD. The statement is false. By the Invertible Matrix Theorem, if Ax=b has at least one solution for each b in R", then matrix A is not
invertible. If A is not invertible, then according to the Invertible Matrix Theorem, the solution is not unique for each b.
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