OK, so should I just leave my answer like the following?
I'm not sure you understood my last post. I was saying that the order you write the matrices matters for when you do the multiplication. You should have
$\displaystyle g' = \left( \begin{array}{cc} 1 & \cos{\theta} \\ 0 & \sin{\theta} \end{array} \right) \left( \begin{array}{cc} 1 & 0 \\ \cos{\theta} & \sin{\theta} \end{array} \right) $
Then carry out the matrix multiplication to get a single 2x2 matrix for g'.