Edit: I can't delete this because it has an answer, so I'll just say that I'm pretty embarrassed by this whole epsidode. I was quite unfamiliar with the site at the time.

OP:

I don't know whether this is meta, or main site, or just not appropriate anywhere so please excuse me. I recently answered this question, using what I thought was my best effort. I went through some other questions that the OP had asked/answered to guage what level we were talking about, and made sure to only talk about things I myself have learned and worked with, so as to ensure correctness. I knew from the beginning that the answer could have been cloudy, but the fact is that the question was also a bit general, and admitted several interpretations. Afterwards I received an anonymous upvote (no comment was left), so I was happy with my work. Of course, after finishing, I was constantly coming up with what if's and possible clarifications, but I decided to wait for the OP's opinion. Today though, my answer was put down as mathematically correct (on first glance), not completely irrelevant — but doesn't answer the question at all." I tried reasoning, but to no avail. So I was left with two options: mentally berate the commenter with all that I could muster, but of course this would get me nowhere, or appeal to a professional community!

Please no one misinterpret, I am not appealing to anyone's sensitivities, as it's not my problem if someone doesn't tactfully treat people. Rather, the comment made me genuinely worried that the answer was indeed as horrific as described, so I am appealing to this community's mathematical knowledge. Hence, my question: Is my answer absurdly off-topic, as well as incorrect in all senses? If so I will delete it. If it's absurdly off-topic, but somewhat correct or at least interesting, then I'll delete it, but save it for revision, in case a question for which it would be on-topic arises.

I really hate adding this part, but situations like these force me to be on the defensive: I am aware that the question has a bounty. I could care less about the bounty, and will be severely offended if we-all-know-what-kind of accusations were directed at me. To anybody (or everybody) to whom this wouldn't have occurred, I apologize. I also apologize if this whole question is out of place (but surely there's some way to ask if an answer is good?)

• The way to appeal to the community's mathematical knowledge is to let the answer sit there and see what happens. – Gerry Myerson Jan 13 '14 at 0:29
• To me that answer is really quite terrible for the given question (I think it might make a great answer for a different question). The reason is that it is very long and as already mentioned, does not answer the very specific question. This could potentially make the asker read carefully through the entire answer, which would take a bit of time, only to end up more confused than when they started, since they would not feel their question had been answered at all. – Tobias Kildetoft Jan 13 '14 at 8:15
• It is not a really bad answer, but it really does not seem to answer the question asked...which is not good. – Pete L. Clark Jan 13 '14 at 13:39
• @GerryMyerson That also came off as a bit aggressive, I don't know if that was your intention. – anon Jan 13 '14 at 15:38
• @TobiasKildetoft All right. I can understand what you guys mean. The only problem is that you seem to know very clearly what the question was asking, and therefore admits a single, clear answer, as if it were something like "how do I do this particular integral", except it wasn't. I really don't see it as this type of question, but I really don't know. As far as I have learned, the determinant is not defined as volume, rather, the definition of volume is a certain determinant, so I really don't know. – anon Jan 13 '14 at 15:43
• There are many ways to define determinant. No aggression was intended; just meaning to bring it to your attention that the site has a built-in mechanism for doing what you are trying to do with this meta-question. – Gerry Myerson Jan 13 '14 at 18:53
• Too many negations in that comment for me to wrap my head around. But being inconsiderate wouldn't be the worst thing I've been accused of, here or elsewhere, so I don't think I'll worry about it. – Gerry Myerson Jan 13 '14 at 19:00
• "So I was left with two options: mentally berate the commenter with all that I could muster, but of course this would get me nowhere, or appeal to a professional community!" You have many more options: Decide that the commenter has a point and edit/delete your question. Decide that the commenter is mistaken and that you will not change their mind, shrug your shoulders and go on to other questions. – Phira Jan 14 '14 at 9:15
• @GPerez "As far as I have learned, the determinant is not defined as volume, rather, the definition of volume is a certain determinant" These are not mutually exclusive facts, but rather points of view. I find the point of view that defines the determinant as oriented volume and then immediately knows why you have this list of properties much more useful than the one you tout as "correct" because you have "learned" it. – Phira Jan 14 '14 at 9:17
• @GPerez The weakness of your answer is the fact that despite its length, the actually relevant part is "So it makes sense that certain properties of how it acts on objects, like the determinant, are preserved.". It does not sound more convincing to me than "So it makes sense that the determinants of the matrices related by an involution on matrices are related by a sign change.". – Phira Jan 14 '14 at 9:20
• @GPerez But really, your answer is not the problem. The problem is that you react to factual critical comments by calling them hostile, aggressive and inconsiderate. – Phira Jan 14 '14 at 9:22
• @GPerez "Points of view" is not the opposite of "factual". You are absolutely wrong that the people who downvote your answer do not share your "point of view that there are more general interpretations than that of euclidean space". Also, I cannot find the place in your comments where you address my example with the sign change. This is the core problem with your answer and you just skip it. I would also like to see your definition of determinant in the context of projective geometry. I do not think that you have a good understanding of these issues. – Phira Jan 15 '14 at 7:17
• @GPerez beyond what Phira described as the problem, I think none of this would've happened if you'd have prefaced your answer with "while not immediately answering your question, of some related interest is the following interpretation..." – Jonathan Y. Jan 20 '14 at 7:36
• The purpose of the meta site is not to ask why a particular answer is good or bad. – kleineg Mar 31 '14 at 16:03
• Checking the discussion thread on the answer this is about, it does look that the answer was justifiably closed. It doesn't answer the question, and doesn't at least explain it offers an alternative view. Yes, having carefully researched work closed sucks. – vonbrand Apr 7 '14 at 19:35

1. Yes, I personally agree with the commenter that this wasn't a great answer. You clearly have a lot of insight into the transpose operation that you wanted to convey, but unfortunately that insight isn't very helpful for understanding why $\det(A^T)=\det(A)$. It doesn't matter how insightful or well-written an answer is if it doesn't sufficiently address the question that was asked.

3. There's not any way to ask whether an answer is good, other than seeing whether it gets upvoted. (In particular, this isn't really what Meta is for, which is why this question currently has downvotes.) Even if an answer receives no upvotes, there's no reason to delete it -- just leave it there on the off chance that it might help someone who reads it. Someone who Googles "matrix transpose duality" or some such might come upon your answer and find it quite illuminating.

I hope that helps!

• Yes, thank you. Also thanks for explaining the downvotes, I don't really know what they accomplish if nobody were to explain themselves. – GPerez Jan 13 '14 at 7:17
• @GPerez: Perhaps you can edit it an "executive summary" at the top of your Answer, to address the specific Question. – hardmath Jan 13 '14 at 12:11
• @hardmath What do you mean exactly? – GPerez Jan 13 '14 at 14:39
• @GPerez: Sometimes when I give a lengthy Answer, I will summarize the conclusion at the top. Calling this an "executive summary" is a bit of a joke, as written business presentations often begin with this as a tactful accomodation for managers who will not make time to read the entire paper. Same idea, give the meat of the response up front, which might motivate a Reader to delve into the details (or decide to opt out). – hardmath Jan 13 '14 at 15:03
• @hardmath Oh okay I get it, I will do so! – GPerez Jan 14 '14 at 15:28

I think you might have misunderstood the comment, given your question here:

Hence, my question: Is my answer absurdly off-topic, as well as incorrect in all senses?

The comment said 3 things:

1. mathematically correct (on first glance)
2. not completely irrelevant
3. doesn't answer the question at all

Of these, both (1) and (2) are positive! (1) means no errors were noticed, and (2) means the answer is related somehow to the question. This is why the third part was introduced with "-- but", because it was a negative point, in contrast to the first two points.

In other words, the comment author was trying to be nice, while pointing out a possible problem with the answer.

No, part (2) of the comment says it is not.

Is my answer incorrect in all senses?

No, part (1) of the comment says that in fact it looks correct.

Since this is Meta, I will also respond to a completely unrelated part of your question:
In another part of your question, you say:

I tried reasoning, but to no avail. So I was left with two options:

But this is not a web site for figuring out who is right or wrong! Rather, it provides crowd-sourced answers to crowd-sourced questions. Crowds on the internet are a rough group and cannot be tamed, especially if they happen to be mathematicians.

Let me be the first to welcome you to math.stackexchange! It takes a little time to get used to the site. Be prepared for comments that will sound impolite to you, and don't waste your mental energy by taking offense or discussing politeness with random people from around the world. This is not a site for making friends or for friendly conversation. It is a site for math. If you stick to the math, you'll find this is actually a very nice site!

• Thank you for your welcoming me! I know what you mean by "a site for math", but this isn't exactly just an encyclopedia is it? What I mean is that there's human interaction, specifically interaction from which one human is supposed to have understood something better. This exchange is typically made optimal when the two, or more, that are interacting all treat each other as equals, is it not? – GPerez Jan 14 '14 at 20:21
• Also, (1) and (2) could hardly be called positive. (1) (on first glance) as in, I'm not going to waste my time reading it a second time to check for full correctness, but I have the feeling that if I do I'd find something wrong. Why only one glance? Isn't this "for math"? If so this is not coherent with the site's purpose. (2) "Not completely irrelevant" means "mostly irrelevant". They could have been positive criticisms, if not for the modifiers, e.g. (1) mathematically correct (2) not irrelevant. Ah but then it almost wouldn't be a criticism, right? the idea wouldn't have gotten across.. – GPerez Jan 14 '14 at 20:29
• ...but this isn't true. The (1) and (2) without the sarcastic modifiers, together with (3) (unaltered) would have made me understand that a suggestion for improvement was taking place, and a nice discussion about what could be improved could have followed. The addition of "(on first glance)" and "completely" make it more than a suggestion for improvement, it becomes a harsh interjection. However, though I admit that when one takes offense, the proceeding responses can have a touch of defensiveness, or even can contain cries for justice or pardon, the impoliteness issue is really not what I.. – GPerez Jan 14 '14 at 20:38
• ...want to discuss! I wanted to discuss, on friendly terms, the actual mathematical question at hand here, the relationship between transpose and dual, and whether this relation can give insight into common properties of the transpose. If you look at all of my comments you'll see that I'm always trying to steer the discussion in this direction, which is what I mean by "...reasoning, but to no avail...". The thing is, nobody can see that this is what I want, because in their eyes I'm just a stuck-up teen who wants to prove himself right above all others. Hence,... – GPerez Jan 14 '14 at 20:48
• @GPerez Meta is not for discussing "the actual mathematical question at hand". – Michael Greinecker Jan 14 '14 at 20:53
• ... "you're not listening", or "tout as correct". I haven't been touting anything as correct and if someone reads through my comments and still thinks so, they are deluding themselves in an effort to keep the image they have already formed of me. I wanted to share my enchantment with new, more general mathematical views and have been met with the response "Case closed. End of discussion." Well that's fine, but certainly not "for math". – GPerez Jan 14 '14 at 20:58
• @MichaelGreinecker Oh alright. I wasn't sure but I thought that it was kind of an in-between. Thanks for clarifying though, does this mean it'll be moved to the main site? In any case I'm sorry it turned out this way, I really did just want to discuss "the actual mathematical question". – GPerez Jan 14 '14 at 21:02
• @GPerez As Gerry Myerson pointed out, there are already mechanisms for judging answers. I want to point out that there is a difference between a question & answer site and a discussion board. MSE (and the whole stackexchange system) is simply not designed to be useful for discussions. – Michael Greinecker Jan 14 '14 at 21:07
• @MichaelGreinecker "I don't know whether this is meta, or main site, or just not appropriate anywhere so please excuse me." This was before, now I'm aware that actively requesting feedback is discouraged. I wasn't sure before though. – GPerez Jan 14 '14 at 21:19
• This answer does seem somewhat disingenuous to me. I wouldn't go so far as GPerez in assigning motivations, but the qualifiers, tone and general phrasing are quite clear; That comment read more as a rebuff than polite criticism. – Jonathan Y. Jan 20 '14 at 7:47
• @Jonathan The comment at hand will indeed appear rude to many people, but not to everyone. People here are all different, some more polite and some more gruff. The comment seems towards the latter type to me, and my personal impression is that the two (weak) "positives" were included precisely to make the comment be less negative. So yes, I really do think the author was intending to make his negative comment be less negative, and that the stated question here is indeed answered by the comment itself, and that stressing over impolite people on the internet is generally a waste of time. – Matt Jan 21 '14 at 21:35
• @Matt, I would probably agree that the comment was thus conceived, but certainly something changed along the way. OP didn't quite hide his displeasure as politely as he might've intended. At this point, I agree with your assessment on both the futility and ultimate lack of need to take matters further by GPerez, but that doesn't mean we should tell him he's imagining things; he wasn't, the 'gruffness' is genuine. That's all I meant by 'disingenuous'. – Jonathan Y. Jan 21 '14 at 22:14
• @Jonathan Then I think we basically agree! Perhaps I should edit the answer to say "(weakly) positive". When I wrote "positive", I was just thinking >0, not >>0. It is completely understandable to me that GPerez found Grigory's comment rude, but also completely understandable to me that Grigory was annoyed by GPerez's answer. I find many people on this site rude, but that's my problem, not theirs. In fact, I value the presence of these people if they contribute interesting mathematics. Mathematics has a history of tolerating individuals with poor social skills. No reason not to do so here too! – Matt Jan 22 '14 at 18:02
• Hear, hear (although, I'd suggest making it $>0-\epsilon$) – Jonathan Y. Jan 22 '14 at 18:14