Is it possible to transform any augmented matrix into RREF?

Question

I have only just started doing RREF problems and are stuck. There doesn't appear to be a way to transform this matrix into RREF. If some augmented matrices cannot be transformed into RREF can you please tell me whether or not the problem I have been presented with can be? The equations are $$ x_1+0+7x_3=17\\ 0+x_2-5x_3=-9 \\ 0+0-13x_3=-30$$

I suspect I may have done a miscalculation so here is the original equations $$ x_1+x_2+2x_3=8\\ -x_1-2x_2+3x_3=1\\ 3x_1-7x_2+4x_3=10$$

Answer 1

By my calculations you should have this:

$$\begin{bmatrix}1&0&7&|&17\\0&1&-5&|&-9\\0&0&-13&|&-26\end{bmatrix}$$

Re-check your work.

Answer 2

here is what my ti-83 gives me::

$$\pmatrix{1&1&2&8\\-1&-2&3&1\\3&-7&4&10}\to \pmatrix{1&0&0&3\\0&1&0&1\\0&0&1&2} $$

so the solution to the original equation is $$x = 3, y = 1, z = 2. $$