# Solution 80: Slaying a Monstrous Determinant using Second Differences

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**Published on Feb 21, 2019**- We have a quadratic sequence embedded within the determinant -- Can we take advantage of this?

This particular problem was inspired by one of the sections on determinant in "Linear Algebra Gems: Assets for Undergraduate Mathematics," a marvelous book I incidentally came upon in a university library, containing many fascinating articles on linear algebra. The article on determinant essentially used the techniques presented in the video, along with a slightly more sophisticated algebra, to derive the formula for a determinant of a square n x n matrix with its terms forming a sequence with (n-1)st common difference. I realized that the formula can be extended to cover the case where each term is written as a product of the aforementioned sequence and a geometric sequence, and the result was the problem in the video. If you are interested in exploring some "non-standard" yet fascinating theorems and ideas from linear algebra, I encourage you to check out the book from the nearest university library. =)

Congratulations to Avi Uday, Benjamin Wang, aby p, Arun Bharadwaj, Andre Ben, Jaleb, 張惟淳, Serengeti Ghasa, Varun Shah, and mstmar for successfully solving this math challenge question! Avi Uday was the first person to solve the question.

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LetsSolveMathProblems3 months ago^{+29}* At 13:33, I should have written 6/80 = 3/40, not 1/15, for the final answer. I apologize for the error. Thank you, Hiren Bavaskar, for notifying me!

JT2 months agoIn solving this problem you took at least 3 side-steps to solve the problem...and these are all excellent and necessary. When I pointed out in an earlier video you became very angry. saying that you don;t explain every point. Well you just did....e.x..p.l..a..i...n e..v..e..r..y. p..o..i..n....t.

Monster Azice3 months ago^{+1}Where does he get these questions? They are super interesting

LetsSolveMathProblems3 months ago^{+2}This particular problem was inspired by one of the sections on determinant in "Linear Algebra Gems: Assets for Undergraduate Mathematics," a marvelous book I incidentally came upon in a university library, containing many fascinating articles on linear algebra. The article on determinant essentially used the techniques presented in the video, along with a slightly more sophisticated algebra, to derive the formula for a determinant of a square n x n matrix with its terms forming a sequence with (n-1)st common difference. I realized that the formula can be extended to cover the case where each term is written as a product of the aforementioned sequence and a geometric sequence, and the result was the problem in the video.

I have authored most of the Weekly Math Challenges so far--The exceptions owe their existence to my younger brother or both of us working in conjunction. I'm glad you find them interesting! =)

Louis Thirion3 months agoYou cannot take the cube root or some other root to reduce a fraktion...

Therefore 216/80^3=6^3/80^3 is not equivalent to 6/80.

The correct answer would be 27/6400.

Sanjay Kannan3 months ago^{+2}The question is to find the cube root of the determinant.

turtlellamacow3 months agoNice problem and very nice explanation as always. I think it would have been a bit cleaner to simply pull all the r's out of the matrix at the beginning (for an overall r^15 outside) rather than carry them along through all the manipulations.

adandap3 months ago^{+2}TBH, in this case I think that simply calculating the determinant first and then using a little algebra wasn't any more difficult than the (admittedly more elegant) solution presented here.

The independence from a is undoubtedly the underlying reason for the very nice cancellations I found - but is there any way to see that from the initial form of the matrix?

Vaidhyanathan A3 months ago@LetsSolveMathProblems how is it to author such beautiful problems,it's really marvellous,answering difficult questions-great,but creating difficult questions is much beyond, 👌👌👌

adandap3 months agoThank you for your response @LetsSolveMathProblems. It begs the obvious generalisation to higher powers - how long will the neat progression save us from an a dependence, or is the step up to cubic enough? I put it into Mathematica to see - the answer is 'no'. Explicitly you get r^15 * 972 (24 + 38 a + 12 a^2 + a^3). Following your reasoning, I thought that a 3x3 with a cubic kind of exhausts that ability of the progression to 'help'. So I generalised it to a 4x4 matrix with cubic terms a^3, (a+1)^3 ... (a+15)^3 etc - and, sure enough, it's independent of a when you do that. But a quartic progression has a determinant which is quartic in a, etc.

LetsSolveMathProblems3 months agoIf the terms of the matrix formed an arithmetico-geometric sequence (instead of a quadratic-geometric sequence, as in the featured problem), it may be intuitively seen that the determinant must be 0 because successive column subtractions will soon yield a 0 column. Perhaps it could be expected that increasing the degree by 1 (from linear to quadratic) would not be quite strong enough to disrupt the determinant's independence from a.

alex zorba3 months ago^{+1}superb!

Angry Tomato!3 months ago^{+10}This is so me at exams, do all hard work good, screw up on easy stuff on end...

Quantum Tlisen3 months ago^{+3}A matrix! That's really nice change of pace in my opinion. Nice video!

shashvat singh3 months ago^{+3}I am in class 10 and i can understand this because the explanation was very nice!

alain Rogez3 months ago^{+1}I tried on my own by hand but it was by far too hard for me.

Hiren Bavaskar3 months ago^{+26}6/80 is not 1/15.. rather it is 3/40 ..please correct it

Hiren Bavaskar3 months ago^{+1}@LetsSolveMathProblems No problem :) Btw I like ur problems.. Keep up the good work!

LetsSolveMathProblems3 months ago^{+4}Oops! Thank you so much for notifying me! I can't believe I made an arithmetic mistake right at the end. I just posted a pinned comment with the correction. =)