Neither did our professor nor book give a proof of this fact, but just in case I need to know that.
6722
Given two square matrices $A$ and $B$, how do you show that $$\det(AB) = \det(A)\det(B)$$ where $\det(\cdot)$ is the determinant of the matrix?
44541
I have looked extensively for a proof on the internet but all of them were too obscure. I would appreciate if someone could lay out a simple proof for this important result. Thank you.
104570
In most of the mechanics textbooks , the  is stated without any justification or an experiment to confirm this law. Also after introducing this law forces are also decomposed without any justification. Please suggest any links or books related to this. Any help will be appreciated. Thanks in advance.
109644
Please go through the following incidents to understand the problem.  Mr. A is a Ph.D. student of some institute who was working with Prof. B of another institute on a very interesting mathematical problem since last three years. Recently, they came across a significant solution to it. They wrote a research article to publish in a very good mathematics journal that accepts the online submission.  Prof. B wanted to submit it to the journal by himself. Mr. A agreed.   A few days ago, Mr. A wanted to know about the submission of their article via telephone. Prof. B informed that the article is submitted. But, Mr. A received any confirmation e-mail neither from the journal nor Prof. B forwarded the submission mail to him. Prof. B did not upload it in arXiv, which is a free preprint repository.  Mathematics journals used to take a long duration to review and publish an article. Prof. B told that submitting this article to arXiv will immediately attract community attention. Many researchers will start working in the new direction. As a result, Mr. A and Prof. B may lose their opportunity to work on that problems. Also, the article may be stolen.  Mr. A did not ask Prof. B to forward the submission confirmation mail to him. Prof. might think that Mr. A is suspecting him. It may affect the relationship. Also, Prof. is a very powerful person in the administration, who may prevent an opportunity for Mr. A to gain a job. Again from some of his friends, Mr. A came to know that Prof. had distributed related research problems to his collaborators.  As a consequence, Mr. A is thinking that Prof. has submitted this work in his own name and he has lost his credit as the first author of this article.  Now the question is: How Mr. A may save his credit and his work without hampering his relationship with Prof B?
43365
I'm not sure how to approach the proof for this. I know this statement is vacuously true, because the empty set does not contain any accumulation points. But I'm having a hard time writing a proof for this. I have looked at other answers on here, however we have not learned what a "ball" is in lecture.
17219
I have just learnt about the chain rule but my book doesn't mention a proof on it. I tried to write a proof myself but can't write it. So can someone please tell me about the proof for the chain rule in elementary terms because I have just started learning calculus.
8454
The question of whether or not a tree that falls in the forest makes a sound - if there is nothing or no one around to hear it - comes up frequently at my house.  So, my question is: is there any way to "prove" or "dis-prove" this using physics? If it can be proven what is the answer?  My idea is yes, of course it makes a sound even if there is nothing to sense it! However, my parents seem to think that if there is nothing to "take in" the sound waves, there is no sounds.
112514
Obviously we can use the completeness of the real numbers (least upper bound axiom, or one of the equivalent principles) to prove the IVT. Can we go in the opposite direction?   This isn't a homework problem or something. I'm just wondering. If the answer is "yes", then I'm not really asking for much explanation. A reference, or a place to look if I'm stuck, will do.
119795
I believe that it does not have to be a greatest element, because to be a greatest element it must be comparable to all the x in X.  I just don't know how to go about proving this.
80417
My book states this as obvious, but it isn't so trivial to me. thanks
41801
Per now, I have basically come upon proofs of the irrationality of $\sqrt{2}$ (and so on) and the proof of the irrationality of $e$. However, both proofs were by contradiction.  When thinking about it, it seems like the definition of irrationality itself demands proofs by contradiction. An irrational number is a number that is not a rational number. It seems then that if we were to find direct irrationality proofs, this would rely on some equivalent definition of irrational numbers, not involving rational numbers themselves.   Are there any irrationality proofs not using contradiction?
18386
I have a professor at my university who uses content from an un-cited textbook that he has never referenced anywhere, including his green sheet (syllabus), as an additional resource. At least some slides he has used include parts of content that have been re-worded and others that are 100% the same. I noticed this by Googling for textbooks on the material we're studying in class and found that examples given in one that I found were exactly the same as what he has provided.  Would this technically be plagiarism? Should he be given the benefit of the doubt in this situation?
2405
I have always taken these kinds of things for granted. Well of course $a(b+c)=ab+ac$! But why? The thought randomly popped in my head, and I realized that I could not prove it.  Perhaps we should take some time to prove these "obvious" things. I feel like if I cannot prove it, then I do not fully understand it. I would really like a proof of this most basic law, because I have been pondering over it for quite a while. Thanks!  Edit: This proof is for real numbers.
82274
I am not sure how to show the second part. Isn’t it obvious from the fact that    G and H are infinite?
113248
My friend asked me this question a few days ago. I felt it's not right but couldn't find a single counterexample. Any comment is appreciated.
32316
So basically this proof centers around proving that (S,*) is a group, as it's quite easy to see that it's abelian as both addition and multiplication are commutative. My issue is finding an identity element, other than 0. Because if 0 is the identity element, then this group won't have inverses.  The set explicitly excludes -1, which I found to be its identity element, which makes going about proving that this is a group mighty difficult.
112593
I just saw the proof that  $$1+2+3+\cdots=-\frac{1}{12}$$ and my brain still hurts.   I completely understood the proof and my question is NOT actually about the proof itself. At the end of the proof, they said that this answer shows itself in quantum physics (e.g. it has a role in why there are 26 dimensions in string theory).  Sadly though, I am no expert in quantum physics and I'm wondering are there any other  physical and/or mathematical situations that this counter intuitive result shows itself?  P.S. Needless to say, it would be great if your answer doesn't need a very deep background in physics or math.