I really don't understand what you did: first, I interchanged the 3rd and 1st rows in the original matrix and brought it to echelon form and got:

Using this matrix a coefficient matrix for a homogeneous system (i.e., equating every row to zero), the general solution is

You can now choose any non-zero value for w and you get a basis for the null space of A. The vector you mention above is NOT in the null space of A, as you can easily check.

Tonio