Ok so I've been set these questions as revision for new course on Algebra and after a summer of no maths my brain has turned to mush!!

For part 1 to 3 I feel reasonably confident I have the correct answers as all pretty simple matrix multiplication but really struggling with part 4.

Is it enough to just show the members of the groups by working out various multiplications of a, a^2, a^3 and b, b^2, b^3, b^4 then doing a multiplication table with all members of group to show that they can be self contained if you like and that each ab,... b^4a^3 etc are members of GL(2,C). This feels a bit laborious and I know how rigorous my lecturer is. just wondering if there is a nicer simpler way to do it without tons of boring matrix multiplication!! And for finding the order I'm at a bit of a loss!!

Any help appreciated!!