I've got two questions which im struggling with:

1. Let $\displaystyle A$ and $\displaystyle B$ be $\displaystyle n\times n$ matrixes such that $\displaystyle A^2=I$,$\displaystyle B^2=I$ and $\displaystyle (AB)^2=I$. Prove that $\displaystyle AB=BA$

2. Let $\displaystyle A$ be some $\displaystyle 2\times2$ matrix such that $\displaystyle AX=XA$ for all real $\displaystyle 2\times2$ matrxes. Prove that $\displaystyle A=\alphaI$ for $\displaystyle \alpha \in R$