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

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 O A Since Ax-bis 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. O B. 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. O 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. 0 is a single vector, and that happens if and only if Ax = 0 has only the trivial solution.

Explanation / Answer

If Ax=0 has a non trivial solution say ,y ie

Ay=0

Then for any x so that Ax=b

A(x+y)=Ax+Ay=b+0=b

x+y is another solution to Ax=b hence, x is not unique

So, Ax=b has unique solution only if Ax=0 has trival solution only

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