Let $A(t)\in SL_r(\mathbb C((t)))$ be a formal power series with matrix coefficients, and let $B(t)\in SL_r(\mathbb C[t])$ and $C(t)\in SL_r(\mathbb C[[t]])$ such that : $$A(t)=B(1/t) \;( ^tA(-t)^{-1})C(t)$$ This implies in particular $$A(t)=B(1/t) \;( ^tB(-1/t)^{-1})A(t)(^tC(-t)^{-1})C(t)$$

Question:

Is it true that we must have $(^tC(-t)^{-1})C(t)=1$ and $B(1/t) \;( ^tB(-1/t)^{-1})=1$?

What can we say about $C(t)$ and $B(t)$? In particular, what is their degree and order respectively?

Is there a description of $A^{-1}(t)$? for example, what is its order? and is some coefficients must be invertible?

Any comment or suggest are welcomed

Thanks.