Suppose Ax = b has a solution. Explain why the solution is unique precisely when

Question

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. 0 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. 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. 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. 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. ? B. ° C. D.

Explanation / Answer

Answer :

Since the system Ax=b is consistent it's 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 .

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