OrangMaths
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- Jan 30, 2023
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I asked SymPy to try your approach, but I can only see patterns in the first and the last polynomial coefficients.What is |A|, when A is 1x1?
What is |A|, when A is 2x2?
What is |A|, when A is 3x3?
Keep doing this until you find a pattern. Then and only then you can try to prove your conjecture by induction.
You missed the 1x1 determinant, lol.I asked SymPy to try your approach, but I can only see patterns in the first and the last polynomial coefficients.
I am posting the results to let smarter people find those patterns:
[math]-x^2+2[/math][math]2 x^{3} - 6 x^{2} + 6[/math][math]- 3 x^{4} + 20 x^{3} - 35 x^{2} + 24[/math][math]4 x^{5} - 45 x^{4} + 170 x^{3} - 225 x^{2} + 120[/math][math]- 5 x^{6} + 84 x^{5} - 525 x^{4} + 1470 x^{3} - 1624 x^{2} + 720[/math][math]6 x^{7} - 140 x^{6} + 1288 x^{5} - 5880 x^{4} + 13538 x^{3} - 13132 x^{2} + 5040[/math][math]- 7 x^{8} + 216 x^{7} - 2730 x^{6} + 18144 x^{5} - 67347 x^{4} + 134568 x^{3} - 118124 x^{2} + 40320[/math][math]8 x^{9} - 315 x^{8} + 5220 x^{7} - 47250 x^{6} + 253092 x^{5} - 807975 x^{4} + 1447360 x^{3} - 1172700 x^{2} + 362880[/math]
The coefficients of the [imath]x^2[/imath] terms i.e. [imath]1, 6, 35, 225, ...[/imath] is the unsigned Stirling numbers of the first kind s(n,3) (OEIS: A000399)[math]-x^2+2[/math][math]2 x^{3} - 6 x^{2} + 6[/math][math]- 3 x^{4} + 20 x^{3} - 35 x^{2} + 24[/math][math]4 x^{5} - 45 x^{4} + 170 x^{3} - 225 x^{2} + 120[/math][math]- 5 x^{6} + 84 x^{5} - 525 x^{4} + 1470 x^{3} - 1624 x^{2} + 720[/math][math]6 x^{7} - 140 x^{6} + 1288 x^{5} - 5880 x^{4} + 13538 x^{3} - 13132 x^{2} + 5040[/math][math]- 7 x^{8} + 216 x^{7} - 2730 x^{6} + 18144 x^{5} - 67347 x^{4} + 134568 x^{3} - 118124 x^{2} + 40320[/math][math]8 x^{9} - 315 x^{8} + 5220 x^{7} - 47250 x^{6} + 253092 x^{5} - 807975 x^{4} + 1447360 x^{3} - 1172700 x^{2} + 362880[/math]
The title of the post asks to use induction.How does "induction" come into play? What technique are you supposed to using for this?
Thank you!
Eliz.
Where did this question come from? I don't think there's a closed form.