in fact 

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as a matter of fact

1 The phrases in fact and as a matter of fact are used to say more about something that you already mentioned; to elaborate on it with something that better illustrates, supports or even proves what you said before

For instance, consider this example:

I absolutely love coffee. As a matter of fact, I have coffee candy in my briefcase for when I can't have an actual cup of coffee.

Here, it has a nuance of «and if you think I'm joking, consider this:...» is used to present a point that shows how serious your are about what you just said — especially when that point occurs to you all of a sudden, while you're talking.

In fact and as a matter of fact are often used after a statement to add some clarification, or simply to interject a point that JUST OCCURRED TO YOU and serves as a perfect example of what you're talking about. For example, In 002 Birdwatching, I use in fact this way. I describe how we consider birdwatchers the epitome of a nerd and then elaborate on that by adding that there's only one thing nerdier — a butterfly catcher. Using in fact in this way has a nuance of «now that I'm thinking/talking about it...».

For some reason, birdwatchers get a bad rap. In movies and shows, if they want to portray someone as a geek, they throw a goofy vest on them, a safari hat, a pair of binoculars, a bird guide and bam!... the epitome of a nerd. In fact, I think the only thing we consider nerdier than a birdwatcher is a butterfly collector, who basically dresses like a birdwatcher with the addition of a butterfly net. 

2 Similarly, in fact and as a matter of fact are normally used at the beginning of a statement when something just occurred to you — a good idea, a last-minute change of plans, or a better alternative to something originally proposed. 

For instance, in the example below, a father is telling his daughter what he and her mother plan to do while in New York, where his daughter lives, when he gets a great idea… why not invite her and her boyfriend to join them for the day?! 

Here, in fact could also be used. However, because the phrase as a matter of fact contains more words, it can be stretched out to sound like you're thinking, pausing, while pondering your new idea, even before you say it. So it makes for a good "filler." For instance, in this example, the father is working out in his head the logistics of having his daughter and her boyfriend join him and his wife even as he's inviting them.

Hey, sweetie, I just wanted to let you know that Phil's wedding is this Sunday. Your mother and I are flying into the city a day early, on Saturday. We figured we’d spend the day shopping, go out and have a nice dinner, catch a show... As a matter of fact, you and Ronnie should join us!

_______________⦿ into the city | New York City is often reduced to the city⦿ to catch a show | to see a (Broadway) musical stage production
Here again, Clara tells Ray not to call her this weekend, but then, thinking better of it, decides she never wants to hear from him again.

Ray had a date with Clara over the weekend and made a complete idiot of himself. He drank too much, flirted with other women, and said some very inappropriate things to Clara that he doesn't even remember.

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Ray: Hey, Clara. It's me, Ray.

Clara: Yes?

Ray: Uh... I just wanted to say I had a nice time on Saturday. Maybe I'll call you again next weekend and see what you're up to?

Clara: Don't call this weekend — I'm going to be out of town. As a matter of fact, don't ever call me again! <click!>

3 The phrase as a matter of fact (but usually not in fact) is commonly used to go along with a suggestion, accept an invitation, agree to something or just say yes in a way that sounds enthusiastic, kind, and grateful. 

It has a nuance of «Since you asked...». For instance, if a stranger at the bank asks if you happen to have a pen they could borrow, answering As a matter of fact, I do sounds kinder than just saying yes. If someone asks you if you'd like something to drink, answering As a matter of fact, I would sounds extremely appreciative.

This t-shirt is cute because it plays to a question that's often used as an ice-breaker when you want to come on to a guy romantically: Do you lift? in the sense of lifting weights because he looks muscular and fit to you. Here, the response is a flattered "Well, yes. As a matter of fact, I do lift. 

Conversely, in the negative, this phrase makes things sounds harsher and snippier than just saying no. Say someone asks you for a cigarette, and you loathe smoking; you could answer As a matter of fact, I don’t which sounds like you’re a bit upset that they even asked you. It sounds much more abrupt and categorical than simply saying no. As such, it’s often used in response to someone asking you if you’re opposed to something in the hopes that you won’t be, and you have to insist that you actually are opposed, as in the first example below.

In this example, Alan wants to be sociable by inviting Connie and her husband to come join them. He checks with Nina in a way that assumes that she won't mind: You don't mind, do you? However, Nina makes it clear that she in fact does mind and won't stand for what he's proposing in a very categorical way.
Nina and Alan are having a drink after a show when they spy an old acquaintance across the bar.____________

Alan: Oh look, it’s Brad and Connie.  I should ask them to join us… you don’t mind, do you?

Nina: As a matter of fact, I do! Or have you forgotten what that alcoholic monster put me through when I worked for her? I swear, Alan, if you call them over here, I’m going to get up and leave. 

4 In certain contexts, as a matter of fact is used to refer to a happy coincidence. For instance, in the example above:

Do you happen to have a pen I could borrow?

— As a matter of fact, I do.

However, this can also be used to respond sarcastically to a question you didn't appreciate or when you feel that someone's asking something that's none of their business, as in the two memes below.

This meme is funny because the little girl is answering the sarcastic question "Was it fun?" in the sense of whether she enjoyed burning down that house. With an evil look, she answers just as sarcastically: As a matter of fact, it was fun... with the corresponding evil gleam in her eye.

This meme is funny because someone asked this guy if he has a retirement plan. He obviously thought it was none of their business, so he answers snidely: As a matter of fact, I do have a retirement plan... FISHING!

Here, Andy's use of this phrase obviously isn't meant to be kind. If anything, he's enthusiastic about the fact that he does have proof — the proof he needs to get David in trouble or even fired. In this usage, it has a (sarcastic) nuance of NOW THAT YOU MENTION IT... Note that in this usage, the verb that follows this phrase is stressed: As a matter of fact, I do!

Andy, who works at a shoe factory, sees the office bully, David, doing something underhanded and goes to the boss. But the boss needs proof. Coincidentally, Andy does have proof and he's more than happy to let the boss know that. 

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Andy: Yesterday, as I was leaving work, I saw David selling some guy shoes — our shoes — out of the trunk of his car.

Boss: That's a strong accusation, Andy. Do you have proof?

Andy: As a matter of fact, I do! I recorded the whole thing with my phone.

How do you say this in your language? 

It may help others if you translate the snippets into your own native language below. 

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