Follow up this matter vs Follow up on this matter

[Winwin2011]

1. I will follow up this matter.
2. I will follow up on this matter.

Are the above both acceptable?

Is it correct to say “ I will address this matter”?

 

[emsr2d2]

1. I will follow up this matter.
2. I will follow up on this matter.

Are the above both acceptable?

Is it correct to say “I will address this matter”?

Note my correction above. Don't leave a space after opening quotation marks.
I find #1 unnatural but #2 OK. Your third option is also grammatically correct. Which one you use depends entirely on context and you haven't given us any.