Suppose $\Lambda$ is a diagonal matrix, and that $AS=S\Lambda$. $ \begin{align*} AS=S\Lambda\\ A^{-1}AS=A^{-1}S\Lambda\\ S=A^{-1}S\Lambda\\ S^{-1}=(A^{-1}S\Lambda)^{-1}\\ S^{-1}=\Lambda^{-1}S^{-1}(A^{-1})^{-1}\\ S^{-1}=\Lambda^{-1} S^{-1}A\\ \end{align*} $
So I got $S^{-1}=\Lambda^{-1} S^{-1}A$.
But if I do it the other way round, $ \begin{align*} AS=S\Lambda\\ ASS^{-1}=S\Lambda S^{-1}\\ A=S\Lambda S^{-1}\\ S^{-1}A=S^{-1}S\Lambda S^{-1}\\ S^{-1}AA^{-1}=\Lambda S^{-1}A^{-1}\\ S^{-1}=\Lambda S^{-1}A^{-1}\\ \end{align*} $ To my surprise, I got $S^{-1}=\Lambda S^{-1}A^{-1}$ ! The $A$ in this is an inverse, which didn't happen in the one above. Weird.
Should $S^{-1}=\Lambda^{-1} S^{-1}A$ or $S^{-1}=\Lambda S^{-1}A^{-1}$ be the correct equation? Why is this like that?