Complete question is;
Suppose Ax = b has a solution. Explain why the solution is unique precisely when Ax = 0 has only the trivial solution. Choose the correct answer.
A. Since Ax = b is inconsistent, its solution set is obtained by translating the solution set of Ax = 0. For Ax = b to be inconsistent, Ax = 0 has only the trivial solution.
B. Since Ax = b is consistent, its solution set is obtained by translating the solution set of Ax = 0. So the solution set of Ax = b is a single vector if and only if the solution set of Ax = 0 is a single vector, and that happens if and only if Ax = 0 has only the trivial solution.
C. Since Ax = b is inconsistent, then the solution set of Ax = 0 is also inconsistent. The solution set of Ax = 0 is inconsistent if and only if Ax = 0 has only the trivial solution.
D. Since Ax = b is consistent, then the solution is unique if and only if there is at least one free variable in the corresponding system of equations. This happens if and only if the equation Ax = 0 has only the trivial solution.
Answer:
Option B: Since Ax = b is consistent, its solution set is obtained by translating the solution set of Ax = 0. So the solution set of Ax = b is a single vector if and only if the solution set of Ax = 0 is a single vector, and that happens if and only if Ax = 0 has only the trivial solution
Step-by-step explanation:
There are different ways of explaining this but we will explain it algebraic ally.
If Ax = b has a solution, then it can be said to be unique if and only if every column of A will be a pivot column. Now, If every column of A will be a pivot column, then it means that there are no free variables, and thus the homogeneous equation will have only the trivial solution.
Also homogeneous equations are always constant.
Thus the correct option is Option B: Since Ax = b is consistent, its solution set is obtained by translating the solution set of Ax = 0. So the solution set of Ax = b is a single vector if and only if the solution set of Ax = 0 is a single vector, and that happens if and only if Ax = 0 has only the trivial solution
