Decomposition of matrix NxN into product of three matrices of size NxN each

Question

Let F be the group of n × n invertible matrices over field X. Let $$G \in F$$
Is there any way to decompose a given matrix of size NxN into product of 3 matrices of size NxN each where I want my middle matrix to be some fixed known matrix. For example - M = AGB where G is already known to me and A,B should belong to F such that $$AB \neq BA$$

Answer 1

I believe one answer could be to take (assuming $G$ non-singular) $$ M=G^{-1}GM. $$ Where we see that $A=G^{-1}$ and $B=M$ (corresponding to your original form $M=AGB$).

This is rather trivial though and I am not sure if this was the answer you are looking for. Maybe indicate any other constraints you might have to result in a less trivial answer?