Answer
No, there does not exist a vector $z$ such that the equation $Ax = z$ has a unique solution.
Work Step by Step
If $Ax =y$ has no solution, then $A$ cannot have a pivot in each row. Thus, because $A$ is a 3 by 3 matrix, it has at most two pivot positions and at least one free variable. Thus, the solution set for $Ax = z$ is either empty or has infinitely many elements.
