(Notational note: Using M for a *vector* will confuse people when doing this kind of straightforward matrix-vector linear algebra arguments. I'd hazard that capital M is universally used to represent a matrix in this context.)
I'm gonna switch away from your M, and use y instead.
You got to here: Let . Then for some .
"Therefore , so is the image, under , of the vector .
Therefore . Therefore ."
That's one half of the proof - the easier half - when doing the proof this way.
Other way begins: "Let ."
You're now trying to show that that implies that .
What you need to do is to find an such that .
You really only know one thing about y - that it's in the image of A. So you're obviously gonna have to use that. Also, you've yet to use that B is invertible, and that's crucial and must be used at some point, because the claim is false otherwise (consider B = 0).
So I've given you: where you are, where you need to go, what you're going to need to immediately use, and what you're going to need to eventually use. With all that in mind, give it a try.