Understanding a proof of Wedderburn's little theorem

Question

I am working on the proof of Wedderburn theorem and I have a problem to understand the part of it. I don't understand why $b_{1}^{-1}a_{1}=\lambda^{i}$ implies the last contradiction $a_{1}b_{1}=b_{1}a_{1}$ and what for the equation $(a^{-1}b)^{r}=1$ was proved? Could anyone elaborate on this, please?

Thank you :)

Source: http://www-fourier.ujf-grenoble.fr/~marin/une_autre_crypto/articles_et_extraits_livres/Herstein-Wedderburn.pdf