fact vs matter of fact

fact and matter of fact both are nouns.

Word	Noun	Adjective	Verb	Adverb
fact	Yes	No	No	No
matter of fact	Yes	No	No	No

As nouns, matter of fact is a hyponym of fact; that is, matter of fact is a word with a more specific, narrower meaning than fact:

fact: a piece of information about circumstances that exist or events that have occurred
matter of fact: a matter that is an actual fact or is demonstrable as a fact

Other hyponyms of fact include case, detail, item, point, particular, specific, general, observation, reason, score, truth.

fact (noun)	matter of fact (noun)
a piece of information about circumstances that exist or events that have occurred	a matter that is an actual fact or is demonstrable as a fact
a concept whose truth can be proved	a disputed factual contention that is generally left for a jury to decide
a statement or assertion of verified information about something that is the case or has happened
an event known to have happened or something known to have existed