# Linear Algebra: A and B are matrices : if $A^2 = I$ and $B^2 = I$ then $(AB)^{-1} = BA$.

I need to prove the following statement or give an example that would prove it wrong. A and B are matrices so that $A^2 = I$ and $B^2 = I$. Is it then true that $(AB)^{-1} = BA$ or if not, can you find an example?

Can i state that $AA=I$ so that A is its own inverse: $A = A^{-1}$ (same for B)? If that's true the proof would of course be easy: $$(AB)^{-1} = B^{-1}A^{-1} = BA$$

• Yes, you got it – user Jan 7 '18 at 21:17

Also, because $$BAAB=B^2=I.$$