Consider the real $2\times2$-matrices that (with respect to the standard basis of $\Bbb{R}^2$ represent rotation around the origin over $120$ degrees and reflection in the $x$-axis. Construct the smallest possible subgroup of $\operatorname{GL}_2(\Bbb{R})$ containing these two matrices. Is the resulting group abelian? Find the order of each element of this group.
How do I have to start? Never got this kind of questions before, the lecturer gave me a terrible explanation and there's nothing to find in the lecture notes.
You start exactly as the question describes; first write down these matrices explicitly. Then look at the subgroup they generate; what do the products of these matrices look like? And their inverses?
While fiddling around, I would advise you to keep in mind the geometric picture of how the matrices act on $\Bbb{R}^2$. That avoids a lot of computations.
EDIT: To make the fiddling around part more concrete; consider an equilateral triangle in $\Bbb{R}^2$ with one vertex on the $x$-axis, and the other two symmetrically above and below the $x$-axis. Now it is easy to verify (both visually and by linear algebra) that the given matrices permute the vertices of the triangle. How many ways are there to permute the vertices in total? And can you get all permutations by applying some combination of these matrices?