This question caught me off guard. I believe I did it the right way, but it was a bit confusing and I wanted a bit more explanation of the process and to see if what I got was accurate.
This is the question:
This is my answer:
$$\vec{x}=\begin{pmatrix}0\\0\\0\end{pmatrix}+t\begin{pmatrix}1\\-2\\1\end{pmatrix}$$
This was my process:
Since this is saying solve the homogenous system, and provides the formula $A\vec{x} = \vec{0}$, that means (matrix)(vector) = (zero-vector) and then I lined up both that way, and ended up doing Gauss-Jordan elimination to get my result and put it in parametric form.
Your answer is correct, and reducing the matrix $A$ is indeed how you should arrive at that answer.
Just a small note, you don't need to write the zero vector. $$\vec{x} = t\begin{bmatrix} 1 \\ -2 \\ 1 \end{bmatrix}$$ is a perfectly correct answer.