Should the matter of fact be hyphenated?

Should the matter of fact be hyphenated?

If the phrase can’t be considered a phrasal adjective, then no. If it can be considered a phrasal adjective, then yes. Examples:

Not a phrasal adjective, therefore not hyphenated:

It appears that, even though it is logically possible for space to be curved, as a matter of fact, it is flat.

So far as the head teacher is concerned, it is a matter of fact that he did exclude him.

As a matter of fact, it is not possible to justify this attack on the basis of international law.

A phrasal adjective, describing a noun, therefore hyphenated:

You're speaking about this in a very calm, matter-of-fact way.

Beautiful scenery, combined with melancholy music and matter-of-fact narration, made a lovely little story.

It is expressed not in high-flown language but as a down-to-earth, matter-of-fact promise.