Complete question is;
Assuming that Ax = b has a solution, explain under what conditions that solution is unique when Ax = 0 only provides the trivial solution. Choose the correct answer.
A. If Ax = b is contradictory, the solutions are derived by shifting the solution set of Ax = 0. Ax = b is inconsistent if Ax = 0 has only the trivial solution.
B. Ax = b being consistent means its solutions result from shifting the solution set of Ax = 0. Hence, Ax = b has a unique solution if and only if its corresponding set yields a single vector. This occurs if and only if Ax = 0 solely has the trivial solution.
C. The inconsistency of Ax = b implies that the Ax = 0 solution set is also inconsistent. Such a solution set is inconsistent if and only if Ax = 0 only contains the trivial solution.
D. If Ax = b is consistent, a unique solution exists only when there is at least one free variable in the corresponding equations. This situation arises only if Ax = 0 provides only the trivial solution.
Answer:
The correct choice is Option B: Ax = b is consistent and its solution set is derived by transforming the solution set of Ax = 0. This means that the solution set for Ax = b is a single vector if and only if the solution set for Ax = 0 is also a single vector, which is only true if Ax = 0 only has the trivial solution.
Step-by-step explanation:
There are various ways to clarify this, yet an algebraic approach will be taken.
If Ax = b has a solution, it is said to be unique if every column of A serves as a pivot column. Therefore, if all columns of A are pivot columns, this indicates that there are no free variables, implying the homogeneous equation will consist of only the trivial solution.
Additionally, solutions of homogeneous equations are consistently constant.
Consequently, the correct choice is Option B: Since Ax = b is consistent, its solution set is achieved by translating the solution set of Ax = 0. Thus, the solution set of Ax = b is a single vector if and only if the solution set for Ax = 0 is singular, which only holds when Ax = 0 possesses the trivial solution.
