Is it possible for a nonhomogeneous system of eight equations in six unknowns to have a unique solution for some right-hand side of constants? Is it possible for such a system to have solution for every right-hand side? Explain. unique Consider the system as Ax = b, where A is a 8x6 matrix. Choose the correct answer below. O A. Yes, No. Since 6 ≤ rank A≤ 8, by the Rank Theorem, dim Nul A=8-rank A≤ 2. Since dim Nul A≤ 2, the system Ax=b will either have no free variables (dim Nul A = 0) or one free variable (dim Nul A = 2). Only for the case dim Nul A=0 will there will be a unique solution for b. OB. No, No. Since A has at most 6 pivot positions, rank A≤ 6. Since rank A≤ 6, Col A will be a proper subspace of R³ and, by the Rank Theorem, dim Nul A≥2. 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 C. Yes, No. Since A has at most 6 pivot positions, rank A ≤ 6. By the Rank Theorem, dim Nul A = 6-rank A≥0. 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 A ≤ 6, Col A will be a proper subspace of R8. Thus, there exists a b in R8 for which the system Ax=b is inconsistent, and the system Ax = b cannot have a unique solution for all b. O D. Yes, Yes. Since A has 6 pivot positions, rank A = 6. By the Rank Theorem, dim Nul A = 6-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.