[Math] RREF of a matrix when the first column is all zeros

Question

I am given the following matrix and asked to convert to row reduced echelon form.
$$
\begin{bmatrix}
0 & 2 & 3 & 0\\
0 & 3 & 10 & 0\\
0 & 0 & -1 & 0\\
\end{bmatrix}
$$

The answer key states that
$$
\begin{bmatrix}
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 0\\
\end{bmatrix}
$$

Is the correct solution. My first instinct was to row reduce so I would get the following
$$
\begin{bmatrix}
0 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
\end{bmatrix}
$$
Why is the above matrix incorrect? Is it because that if there is any row with all zeros it must be at the bottom for the matrix to be in RREF?

Answer 1

All zero rows of a matrix must be at the bottom for a matrix to be considered in Row Reduced Echelon Form.