questions below:

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 solution for every righthand side? Explain. Consider the system as Ax = b, where A is a 7 x 4 matrix. Choose the correct answer below. OA. OB. OC. OD. 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. Yes, No. Since 4 5 rank A S 7, by the Rank Theorem, dim Nul A = 7 - rank AS 3. 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. No, No. Since A has at most 4 pivot positions, rank A S 4. Since rank A S 4, Col Awill be a proper subspace of [R7 and, by the Rank Theorem, dim Nul A2 3. Thus, for any b, there will exist either innitely many solutions, or no solution. So, Ax = b cannot have a unique solution for any h. Yes, No. Since A has at most 4 pivot positions, rank A S 4. By the Rank Theorem, dim Nul A = 4 - rank A2 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 S 4, Col A will be a proper subspace of R7. Thus, there exists a b in [R7 for which the system Ax = b is inconsistent, and the system Ax = b cannot have a unique solution for all b. 1 4 -1 - 16 8 If A= and AB = , determine the first and second columns of B. Let b, be column 1 of B and b2 be column 2 of B. 4 3 -9 -16 . . . by b, =Find an LU factorization of the matrix A (with L unit lower triangular). 5 5 A= -4 - 3 L= U=