Allowed row operations on matrices

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 1

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.

Answer 2

No you can not subtract a row from its self.

Please see this link.

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.

Answer 3

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 $(0,0,0)$.