I need to find $r, s \in \mathbb{R}$ such that the following system $A\textbf{x}=\omega$ is true for all $x, y \in \mathbb{R}$, with $\textbf{x}=(x,y)$. I let $A$ be the coefficient matrix and e $\omega \in \mathbb{R}^2$ a vector, then
$$ A=\left(\begin{array}{cc} 2-2s & 2 r \\ 2 r & 2s-2 \end{array}\right), \quad \omega=\left(\begin{array}{c} s-1 \\ -r \end{array}\right) . $$
For the system to be true for any $x$ and $y$ I need it to be compatible and have $\infty^2$ solutions: hence I need the rank of the augmented matrix $(A|\omega)$ and the matrix $A$ to be both equal to 0. Doing the computations I get $r=0$ and $s=1$. I am asking for a verification of my reasoning and my result. Thank you very much!