When you have the system

and you are solving for

.

The matrix you end up with is actually:

Now if you done the row-reductions (assuming you were correct!) we get:

You can bring the bottom three rows to:

However this matrix is still now in row-reduced echelon form.

We will add the (3rd row

2) to 2nd row and 1st row.

Finally, add (2nd row

) to 1st row:

This tells you that:

.

Thus,

(notice that

is "free").

This means if

then

solves

, and conversely.

Thus, the nullspace is

We see the dimension is one, and so the nullity is one.