I've always been curious about this: What is the grammatical reason for 的 at the end of these short responses?
Q: 這樣好嗎？ A: 好的。
Q: 你是美國人嗎？ A: 是的。
Q: 你會彈鋼琴嗎？ A: 會的。
Can someone shed some light on it for me?
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The function of
的 is well stated in 漢典:
4 . 助词，用在句末， 表示肯定的语气 ，常与“是”相应：这句话是很对～。
At the end of a sentence, 的 is an auxiliary word, indicating a response with emphasized confidence. It is often used in the "是……的" structure.
I think it's not proper to literally translate
表示肯定的语气 into "indicating a positive response". That's because, although the example sentence
这句话是很对的。 This sentence is quite right.
is a positive response, a negative response in the 是...的 structure perfectly make sense too!
这句话是不对的。 This sentence is not right.
The 是...的 structure has been well studied in this question. See golddc and 孤影萍踪's answer.
Come back to OP's question. Indeed, when 的 is used in a short answer, it's generally a positive answer.
Q: 你會彈鋼琴嗎？ A: 會的。
Q: Can you play piano? A: (Yes), I can.
But I would rather say, this effect is related to psychology. Most of the time, we tend to be confident to give a positive response; however on the contrary, a negative response is seldom stated with confidence.
Q: 你會彈鋼琴嗎？ A: 不會的。
The grammar is perfect in this sentence, but it would sound like:
Q: Can you play piano? A: Of course I can't.
You would feel such an answer sounds cynic.
For a neutral question (I mean, whether a "yes" or a "no" answer won't be specially expected), a negative response with confidence is commonly used too.
Q: Will he come? A: I'm sure he won't.
Q: Are you a party member? A: Of course not.
It is grammatically perfect and sounds right.
肯定的语气 is a little ambiguous. It can mean either "positive mood" or "confident mood". In this auxiliary word 的 case,
肯定的语气 is better to be considered as the confident mood.
I think there is the grammatical and dictionary version of the explanation, but also you can look at it from the way people talk in everyday life. In my experience, omitting 的 at the end of a response seems to cut the response short, as if someone is either given a short response, like 會 (yep) versus 會的 (I can/I know); 是 (yes) versus 是的 (that's correct). The other way to look at it is to express a more affirmative mood as described above by Stan.
的 has different meanings, but in your examples, all three 的 are auxiliary words. In general, it should be located at the end of sentence, indicating a positive response.
For more info, please check online Chinese dict 汉典: http://www.zdic.net/z/1f/js/7684.htm