For which values of $a, b$ does the system of equations not have any solutions? I am trying to solve the following problem:

For which values of $a$ and $b$ does the linear system represented by the augmented matrix not have any solution?
$$ \left[\begin{array}{ccc|c}
      1&-2&3&-4\\
      2&1&1&2\\
      1&a&2&-b
     \end{array} \right] $$


Truthfully, I don't know where to start. Thus, any help is welcomed. Thank you very much!
EDIT: I tried to solve the problem on my own using Gaussian elimination, but I am not sure that the solution is the right one, or is that the right way to go.
EDIT2: OK, here is how I tried to solve it using Gaussian elimination:
multiplication of the third row with -1, and adding with the first row, the I got
$$ \left[
    \begin{array}{ccc|c}
      1&-2&3&-4\\
      2&1&1&2\\
      0&2a&-2&2b
    \end{array}
\right] $$
Dividing the third row with 2:
$$ \left[
    \begin{array}{ccc|c}
      1&-2&3&-4\\
      2&1&1&2\\
      0&a&-1&b
    \end{array}
\right] $$
Now my problem is, that I have two variables in one equation. I have no idea how to go next.

By row reduction the system becomes (if I didnt make a mistake (highly likely))
$$\begin{pmatrix}
1&0&1&0\\
0&1&-1&2\\
0&0&a+1&-b-2a\\
\end{pmatrix}$$
In order for the rank to be less than $3$ we need that $a+1=0$, so $a=-1$ for no solution we then need $-b-2a \neq 0$ so $b\neq 2$
