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Let a $3 \times 3$ matrix have the elements $1,2,\dots,9$. What is the maximum value the determinant may have?

I have found the desired value and an intuitive feeling/approach as to why that must be optimal. I struggle to really prove that claim, though.

Qi Zhu
  • 7,999
  • What's the value and what's your approach? – JKEG Mar 24 '18 at 23:21
  • Good question. I'm interested in the minimum value too. – ChoF Mar 24 '18 at 23:24
  • The value ist $412$. My approach, well, there is a "positive part" and a "negative part" of the sum. By rearrangement inequality, we can see that the positive part is maximized with $9 \cdot 8 \cdot 7 + 6 \cdot 5 \cdot 4 + 3 \cdot 2 \cdot 1$. With a heuristic approach, some intuition and rearrangement, one gains a minimal expression for this arrangement. By intuition, this should be optimal. Apparently, it also is.

    I didn't include that in the initial post to not spoil interested people.

     – Qi Zhu Mar 24 '18 at 23:24
  • Minimum positive value would be $1$. As for minimum value, shouldn't that just be the negative of the maximum value by symmetry? – Qi Zhu Mar 24 '18 at 23:25
  • @Kezer You are right. If the max value is found, the negative must be its minus value because interchanging two rows multiplies its determinant by $-1$. So it makes sense to consider the minimum positive value, and you said it is $1$. I'd like to see your minimum positive answer too. – ChoF Mar 24 '18 at 23:47
  • @ChoF I just saw that I miscalculated, I thought I got $1$ but instead the answer in that determinant is $0$, sorry, my fault! That makes the question of a minimum positive value interesting - I assume it's even more difficult to find than the maximum value.

    As for my answer, above I've posted the approach, what do you want to be clearer? Again, I only have an intuitive approach and not a proof, that's why I'm asking on this forum.

     – Qi Zhu Mar 25 '18 at 00:00
  • Duplicate of https://math.stackexchange.com/questions/1465627/maximising-determinant-problem – Gerry Myerson Jun 01 '21 at 09:31

1 Answers1

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At the Online Encyclopedia of Integer Sequences the maximal value is tabulated for $n\times n$ matrices for $n\le7$. Bounds are given for $8\le n\le10$, and an upper bound good for all $n$ is given. A reference and a link is given to Ortwin Gasper, Hugo Pfoertner and Markus Sigg, An Upper Bound for the Determinant of a Matrix with given Entry Sum and Square Sum, JIPAM, Journal of Inequalities in Pure and Applied Mathematics, Volume 10, Issue 3, Article 63, 2008.

412 is correct for $n=3$.

Gerry Myerson
  • 179,216
  • Thanks for the answer. Do you have any idea on the minimum positive determinant? I guess that it is 1 for $n\geq3$. – ChoF Mar 25 '18 at 00:58
  • I would also guess it's 1. – Gerry Myerson Mar 25 '18 at 02:42
  • That's very interesting, thanks. :) As for the minimum positive determinant, I guess for small $n$ it's at least easy to write code to find the answer (possibly giving a good conjecture)? – Qi Zhu Mar 25 '18 at 15:38