linear algebra - Row Equivalent Matrices

If I have a matrix $A$, where there are zeros everywhere apart from the first row, what are the matrices that are not row equivalent to $A$.

I know that if two matrices are row equivalent, we can get from one to the other using elementary row operations.

My idea so far is that it would be all the matrices with $1$ or more zeroes in a row, but not all zeroes in the same row. I'm unsure how to put this in to a formal argument.

Answer

Equivalently, two matrices are row equivalent if they have the same
row space. The matrix you describe can be written

$$
A=\left(\begin{array}{c}
a\\
0\\
\vdots\\
0
\end{array}\right)
$$
where $a$ is a row vector. The row space of $A$ is all vectors of
the form $\alpha a$, where $\alpha$ is a (possibly zero) constant.