Need a help with the first order linear vector recurrence relation.

Question

I have a sequence of vectors $\{\psi_{k}\}_{k\in\mathbb{Z}}$ where $\psi_{k}\in{\mathbb{C}}^{2}$ and they obey the following linear recurrence relation $$\psi_{n}=c_{n-1}\psi_{n-1}$$ Where $$c_{k}\in{M_{2\times{2}}(\mathbb{C})}$$ I had the following idea. We multiply the relation by $$\prod_{k=-\infty}^{n-1}c_{k}^{-1}$$ To give $$\prod_{k=-\infty}^{n-1}c_{k}^{-1}\psi_{n}=\prod_{k=-\infty}^{n-2}c_{k}^{-1}\psi_{n-1}$$ Then we call $$\chi_{n}=\prod_{k=-\infty}^{n-1}c_{k}^{-1}\psi_{n}$$ So, the reccurence becomes $$\chi_{n}=\chi_{n-1}$$ Hence $$\chi_{n}=\chi, \ \forall{n}$$ Where $\chi$ is some constant vector. So, that $$\psi_{n}=\Big(\prod_{k=-\infty}^{n-1}c_{k}^{-1}\Big)^{-1}\chi$$ Can anyone give it a sober look, and, perhaps, find some mistakes. Pre-thanks!!!