how to prove matrix AB=BA if A^2=B^2=identity

Question

anonymous · Answer

Can we assume:
$$
A^{-1}A=AA^{-1}=e
$$Or shall I prove that?

anonymous · Answer

It's commutativity of the inverse and associativity of the operation. Not a terrible proof.

anonymous · Answer

There's an even simpler proof:

We wish to show
$$
AB=BA
$$Note that:
$$
A=A^{-1}
$$And:
$$B=B^{-1}$$

We have:
$$
AB(AB)^{-1}=AB(A^{-1}B^{-1})=A(BA^{-1})B^{-1}=e
$$Then:
$$
A^{-1}A(BA^{-1})B^{-1}=A^{-1}=(BA^{-1})B^{-1}
$$And:
$$
(BA^{-1})B^{-1}B=A^{-1}B=BA^{-1}e=BA^{-1}
$$So, we have:
$$
A^{-1}B=AB=BA^{-1}=BA
$$Which is what we wished to prove.