Determinant and Inverse of a predefined matrix. The question and bounty are updated.
See the attachment. where (a, b) = gcd(a, b), mu = Mobius functions, phi = Euler function.
I have agreed to increase the bounty as the solver requested to: $75 + $75 = $150.
The solver has agreed to provide detailed proofs for both i) and ii) as follows:
i) Proving that Det(C) is not 0 for any n ≥ 3.
ii) Finding the inverse of C for any n ≥ 3, and then proving that the inverse is correct.
Gestyue
21
Answer
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Alessandro Iraci
1.7K
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Hi Alessandro, I'm too busy on research, I may come to review it after some time. Thanks!
The answer is accepted.
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What are mu and phi?
Mobius functions, Euler function
One needs to spend time analyzing the question for the first few values of n to develop an intuition and determine if a general solution exists. I would say this is a very time-consuming and challenging question that deserves a higher bounty.
OK, I will increase bounty.
Hello, I found an explicit formula for the determinant and the inverse. It's not too bad. Numerical computation confirms that it actually is the inverse. This alone took quite a bit of work: I can provide you with the formula for the given bounty, or I can try to write down a detailed proof, but that would require more work (and thus, a higher bounty). I don't expect it to be too complicated if you want to try yourself. Let me know if either option works for you!
Hello, Have you proved both the determinant and the inverse? I have proof but too complicate and truly believe a simpler proof exists. Just let me know what is the reasonable bounty if you write up the detailed proofs for both.
I didn't try to write down a proof yet. I have a formula and I expect it to just be a matter of computing things correctly, but it's probably some hours worth of work. The determinant will follow immediately from the formula for the inverse. I'd say twice this bounty would be reasonable, but at this point I already put in most of the work, so I will take the question anyway. Just put a price you would be comfortable with.
Hi, I just want to make sure we're on the same page: I need the proof, not just the correct computations. Before we can be certain that the determinant is non-zero, how can we derive the formula for the inverse? I just want to ensure we're aligned. Once I feel we're on the same page, I will consider your bounty request.
Well if I give you a matrix, and show that your matrix multiplied by the one I provide gives the identity matrix, then the determinant must be non-zero. The determinant of the inverse is actually very easy to compute, so from that you can also derive the original determinant. I can give you the explicit inverse (including the determinant) for the current bounty, or include a proof that the given matrix is actually the inverse, for extra compensation. Even as separate questions, if you prefer.
I'm asking for extra because I think the proof is not easy. I'm certain that my formula is correct (I checked it by computer up to n=100 and it works), but completing the proof is more challenging than I expected. I think I can do it, but it's going to take me a while.
Hi, we may look at the problem from a different angle. If matrix A and matrix B are matrices with numerical values, then if AB = the identity matrix, B is the inverse of A. But if A and B are matrices with formulaic characters, we cannot say that. If matrix C = {{a, b}, {c, d}}, and D = {{d/(ad-bc), -b/(ad-bc)}, {-c/(ad-bc), a/(ad-bc)}}, and CD = the identity matrix, we cannot say D is the inverse of C, because to say D is the inverse of C, Det(C)=(ad-bc) must be non-zero
Maybe I wasn't clear. I have an explicit formula for the inverse. I can write it as a product of triangular matrices with actual numbers on the diagonal. I'm never dividing by anything, so there is no issue with that. The fact that the determinant is non-zero follows as a corollary.
As a similar example: say you have the matrix [[cos x, sin x] [-sin x, cos x]]. Then I can just say that the inverse is [[cos x, -sin x] [sin x, cos x]] and show that the product is the identity. The determinant will automatically be non-zero, no need to check it in advance.
I looked at my post and realized: I would need help with: i) Proving that Det(C) is not 0 for any n ≥ 3. ii) Finding the inverse of C for any n ≥ 3. For i), I made it clear that I need a proof. For ii), I may not have made it clear (even though I meant to) that a proof is also required. So, could you provide detailed proofs for both i) and ii)? If so, we can discuss increasing the reasonable bounty. I want to make sure both you and I are comfortable. Thanks!
If you want a proof, I can give you a proof, but it will take me time to get one. If that helps, I'm a university professor in mathematics, I taught linear algebra multiple times, I know what I'm doing. What I meant is that, if a formula is good enough, then the current bounty is good enough. If you want a proof, it will take me more time, so for that I would ask for a higher bounty. Please notice that your question is research-level, it's not basic algebra.
I think I got a proof, bar a few details. The estimate bounty I gave you is probably even low for the amount of work, but again, at this point I have already put in the effort. Give me a fair price and I'll post the proof (but I need time to polish it first).
The determinant equals (-1)^(n-1) * 2^[n/2] * n!, where [.]=floor, cf. oeis.org/A268363 .
and the inverse is always the transpose of the comatrix divided by the determinant. So there's an obvious explicit formula for the inverse, it's just needed to prove the formula for the determinant, e.g. by induction. That may be tedious, though.
Yeah that's the determinant. It's very obvious from my formula - which is NOT the transpose of the comatrix, it's very explicit. I have the answer OP wants. Proving the formula for the determinant by induction is close to impossible in my opinion. The matrix does not have a simple structure. I have a factorization involving the properties of mu and phi, which makes everything clear. Took me close to 8 hours though.
Hi Alessandro, Thank you for the information. I'm willing to increase the bounty as you requested to: $75 + $75 = $150, if you provide detailed proofs for both i) and ii) as follows: i) Proving that Det(C) is not 0 for any n ≥ 3. ii) Finding the inverse of C for any n ≥ 3, and then proving that the inverse is correct. Are you comfortable with this? If so, please let me know how to ensure you are the one who takes the question. Thanks!
That's fine with me! I think you can assign a question directly, but I'm not sure. Otherwise you can just ask people not to take it.
Hi Alessandro, I don't know how to assign the question directly to you. Is there something you can do to take ownership of the question?
Just update the bounty, and I'll take it.
Hi Alessandro, Done.
Alright, taken. I'll typeset the answer carefully and send it when I'm done. Might take a while.
Thanks Alessandro!