gauss jordan elimination

So you have
$\left(\begin{array}{c}x' \\ y'\end{array}\right)= \left(\begin{array}{cc}cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{array}\right)\left(\begin{array}{c } x \\ y\end{array}\right)$

The augmented matrix is
$\left(\begin{array}{ccc}cos(\theta) & -sin(\theta) & x' \\ sin(\theta) & cos(\theta) & y'\end{array}\right)$
Do you know how to row-reduce that?

This is, by the way, the matrix corresponding to rotation through angle [itex]\theta[/itex]. It has an obvious, easy, inverse that you can use to check your answer.