Implicit function theorem for matrices

Question

I have a homework assignment for the question below and
I honestly don't even know where to start. 
Any hints or solutions would be very helpful. 
Thanks!

Question:

Let $M$ denote the set of complex $n \times n$ - matrices endowed with the operator norm.

For $A$, $B$ and $C$ in $M$, consider the equation:

$ABCA + CABC = BCAB + I$

Use the implicit function theorem to prove that there exists a smooth function $F$, defined in some neighborhood of $(I,I)$ such that,

$F(I,I)= \mathbf{I}$ and such that $C = F(A,B)$ satisfies the above equation for $A$ and $B$ close enough to $I$.

Answer 1

So you have it all set up: Consider the function $$f\colon M\times M\times M \to M, \quad f(A,B,C) = ABCA+CABC-BCAB.$$ We need to see that $\dfrac{\partial f}{\partial C}(I,I,I)$ gives an invertible map $M\to M$. As a hint, consider $f(I,I,I+tC')=(I+tC')^2$.

EDIT: Then we have $$\frac{\partial f}{\partial C}(I,I,I)(0,0,C') = 2C',$$ so $\dfrac{\partial f}{\partial C}(I,I,I)$ is an invertible linear map from $M$ to $M$.