Is it possible for a nonhomogeneous system of seven equations in four unknowns to have a unique solution for some right-hand side of constants? Is it possible for such a system to have a unique solutio for every right-hand side? Explain. Consider the system as Ax = b, where A is a 7x4 matrix. Choose the correct answer below. O A. Yes, No. Since A has at most 4 pivot positions, rank AS4. By the Rank Theorem, dim Nul A = 4 - rank A20. If dim Nul A = 0, then the system Ax = b will have no free variables. The solution to Ax = b, if it exists, would thus have to be unique. Since rank As 4, Col A will be a proper subspace of R'. Thus, there exists a b in R' for which the system Ax =b is inconsistent, and the system Ax = b cannot have a unique solution for all b. O B. Yes, Yes. Since A has 4 pivot positions, rank A = 4. By the Rank Theorem, dim NulA = 4 - rank A = 0. Since dim Nul A = 0, the system Ax = b will have no free variables. The solution to Ax = b, if it exists, would thus have to be unique for all b. O C. No, No. Since A has at most 4 pivot positions, rank AS4. Since rank As4, Col A will be a proper subspace of R' and, by the Rank Theorem, dim Nul A2 3. Thus, for any b, there will exist either infinitely many solutions, or no solution. So, Ax = b cannot have a unique solution for any b. O D. Yes, No. Since 4s rank As7, by the Rank Theorem, dim Nul A = 7- rank As3. Since dim Nul A s 3, the system Ax = b will either have no free variables (dim Nul A = 0) or one free variable (dim Nul A = 3). Only for the case dim Nul A = 0 will there will be a unique solution for b.
Is it possible for a nonhomogeneous system of seven equations in four unknowns to have a unique solution for some right-hand side of constants? Is it possible for such a system to have a unique solutio for every right-hand side? Explain. Consider the system as Ax = b, where A is a 7x4 matrix. Choose the correct answer below. O A. Yes, No. Since A has at most 4 pivot positions, rank AS4. By the Rank Theorem, dim Nul A = 4 - rank A20. If dim Nul A = 0, then the system Ax = b will have no free variables. The solution to Ax = b, if it exists, would thus have to be unique. Since rank As 4, Col A will be a proper subspace of R'. Thus, there exists a b in R' for which the system Ax =b is inconsistent, and the system Ax = b cannot have a unique solution for all b. O B. Yes, Yes. Since A has 4 pivot positions, rank A = 4. By the Rank Theorem, dim NulA = 4 - rank A = 0. Since dim Nul A = 0, the system Ax = b will have no free variables. The solution to Ax = b, if it exists, would thus have to be unique for all b. O C. No, No. Since A has at most 4 pivot positions, rank AS4. Since rank As4, Col A will be a proper subspace of R' and, by the Rank Theorem, dim Nul A2 3. Thus, for any b, there will exist either infinitely many solutions, or no solution. So, Ax = b cannot have a unique solution for any b. O D. Yes, No. Since 4s rank As7, by the Rank Theorem, dim Nul A = 7- rank As3. Since dim Nul A s 3, the system Ax = b will either have no free variables (dim Nul A = 0) or one free variable (dim Nul A = 3). Only for the case dim Nul A = 0 will there will be a unique solution for b.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Is it possible for a nonhomogeneous system of seven equations in four unknowns to have
for every right-hand side? Explain.
unique solution for some right-hand side of constants? Is it possible for such a system to have a unique solution
Consider the system as Ax =b, where A is a 7x4 matrix. Choose the correct answer below.
O A. Yes, No. Since A has at most 4 pivot positions, rank AS4. By the Rank Theorem, dim Nul A = 4 - rank A20. If dim Nul A = 0, then the system Ax = b will have no free variables. The solution to
Ax = b, if it exists, would thus have to be unique. Since rank As4, Col A will be a proper subspace of R'. Thus, there exists a b in R' for which the system Ax = b is inconsistent, and the system
Ax = b cannot have a unique solution for all b.
O B. Yes, Yes. Since A has 4 pivot positions, rank A = 4. By the Rank Theorem, dim Nul A = 4- rank A= 0. Since dim Nul A = 0, the system Ax = b will have no free variables. The solution to Ax = b, if it
exists, would thus have to be unique for all b.
O C. No, No. Since A has at most 4 pivot positions, rank AS4. Since rank As4, Col A will be a proper subspace of R' and, by the Rank Theorem, dim Nul A23. Thus, for any b, there will exist either
infinitely many solutions, or no solution. So, Ax = b cannot have a unique solution for any b.
O D. Yes, No. Since 4s rank As7, by the Rank Theorem, dim Nul A =7- rank As 3. Since dim Nul A s3, the system Ax = b will either have no free variables (dim Nul A = 0) or one free variable
(dim Nul A = 3). Only for the case dim Nul A = 0 will there will be a unique solution for b.
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