Given the system of equations
$$x-y+2z=6$$
$$x-ay-az=0$$
$$-ax+y+7z=b$$
I'm asked to find the values of $a$ and $b$ for which the system
$$A\begin{pmatrix} x\\ y\\ z \end{pmatrix} = \begin{pmatrix} 6\\ 0\\ b \end{pmatrix}$$
is independent, as well as for when $(0,-2,2)$ is the only solution. The natural way forward would be to find the rref of the corresponding matrix, the issue is that I end up getting some complicated rational functions of $a$ and $b$, so I wonder if there is a simpler way of solving the excercise.
I've also thought of computing the determinant and determining when it is $0$. However, the inverse of $A$ also involves some complicated rational functions. Is there a simpler way of tackling the problem?