Need help calculating this determinant using induction

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.
 
Steven G said:
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.
Click to expand...
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]
 
blamocur said:
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]
Click to expand...
You missed the 1x1 determinant, lol.
I didn't expect that I would see a pattern but that is what the OP has to look at. I can't think of another way.
I will look at what you posted closely to see if I can see a pattern.
 
I suspect that there is more to this problem. If x=0, then the answer is obvious. If x = # of rows in A, then the answer is obtainable. I bet x=n, where A = nxn
 
blamocur said:
[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]
Click to expand...
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)
where [imath]s(n, k) = k \cdot s(n-1, k) + s(n-1, k-1)[/imath]
[math]n=0: s(0,3) = 0\\ n=1: s(1,3) = 0\\ n=2: s(2,3) = 0\\ n=3: s(3,3) = 1\\ n=4: s(4,3) = 6\\ n=5: s(5,3) = 35\\ n=6: s(6,3) = 225\\[/math]
The value s(n,3) is the number of ways to partition a set of n elements into 3 non-empty subsets.
 
How does "induction" come into play? What technique are you supposed to be using for this?

Thank you!

Eliz.
 
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