Currently doing this question and I've just finished the first "prove that" bit, so I thought I'd come here and read your bit on it. I had something slightly similar:
1. It's obviously a rotation so let's see if I can work with that. Hmmm, well, rotating m-1 anticlockwise is the same as rotating m clockwise (you know what I mean), could I pair these terms up pairwise? Is there some condition on m being even/odd only? *check* nope, damn. I'll have to split casewise... this is getting too long, let me give up this approach.
EDIT: It works now
2. Ah! They want induction, obvioooously. *starts writing out induction* lol nopes this isn't gonna work.
3. Okay, so... what series starts out with a number and then keeps that number constant whilst increasing the power. Is this the exponential series? Naaah, you cray m8, no factorials. AHHH!!!! It's a geometric series you retard.
Okay, so:
Unparseable latex formula:\displaystyle[br]\begin{equation*} \mathbf{I} + \mathbf{M} + \mathbf{M}^2 + \cdots + \mathbf{M}^{m-1} = \mathbf{I}\left(\mathrbf{I} - \mathbf{M}^m\right)(\mathbf{I}-\mathbf{M})^{-1} \end{equation*}
.
Since
M=I then
I−Mm=O as required.