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 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).
No, because row operations on a matrix $A$ preserve the solutions of $Ax=0$.

For the first matrix, the system is $x\_1+x\_2=0$, $x\_3=0$, which has general solution $(t,-t,0)$.

For the second matrix (the identity matrix), the general solution is $(0,0,0)$.