My prof taught us that during Gaussian Elimination, we can perform three elementary operations to transform the matrix:
1) Multiple both sides of a row by a non-zero constant
2) Add or subtract rows
3) Interchanging rows
In addition to those, why isn't removing zero rows an elementary operation? It doesn't affect the system in any way. Define zero rows to be a row with no leading variables.
For example isn't $\begin{bmatrix}a & b & k\\c & d & m\end{bmatrix} \rightarrow
\begin{bmatrix}a & b & k\\c & d & m\\0 & 0 & 0\end{bmatrix}$
Best Answer
Here is a slightly more long-range answer: a matrix corresponds to a linear operator $T:V\to W$ where $V$ and $W$ are vector spaces with some chosen bases. The elementary row operations (or elementary column operations) then correspond to changing the basis of $W$ or of $V$ to give an equivalent matrix: one which represents the same linear operator but with the bases changed around. Under this correspondence you can get all the possible matrices corresponding to the linear operator $T$ by doing elementary row and column operations.
Adding or removing a row of zeroes will not give you a matrix corresponding to the linear operator $T$.