Question: Can I get this matrix

\begin{bmatrix}
1&1&0\\ 0&0&1\\ 0&0&0
\end{bmatrix}

from the identity matrix $I\_3$ like this:

\begin{bmatrix}
1&0&0\\ 0&1&0\\ 0&0&1
\end{bmatrix}

Add second row to first row. Then subtract second row from itself. Is the subtraction of a row from itself allowed?

Answer: 
No, you cannot subtract a row from itself!

Besides, your first matrix is *not* invertible, whereas the identity matrix is. 

Valid row operations on a singular matrix (non-invertible) will *never* yield a non-singular (invertible) matrix, while valid row operations on an invertible matrix will always yield another invertible matrix.
No you can not subtract a row from its self.

Please see this [link](http://en.wikipedia.org/wiki/Elementary_matrix#Operations).

The matrix you want to obtain is a rank 2 matrix where as $I\_3$ is a rank 3 matrix.
In other words as amWhy suggested you can not get a non invertible matrix from an invertible matrix through valid [elementary row operations](http://en.wikipedia.org/wiki/Elementary_matrix#Operations).