You are Here: Home >< Maths

Determinants problem watch

1. I'm trying to come up with a conjecture about the value of the determinants of matrices W[n]. Problem being that the determinants are horrible polynomials:

detW[1] := 1
detW[2] := x - 1
detW[3] := x (x + 1) (x - 1)^3
detW[4] := x^4 (x - 1)^6 (x + 1)^2 (x^2 + x + 1)
detW[5] := x^10 (x^2 + 1) (x^2 + x + 1)^2 (x + 1)^4 (x - 1)^10
detW[6] := x^20 (x^4 + x^3 + x^2 + x + 1) (x^2 + 1)^2 (x^2 + x + 1)^3 (x + 1)^6 (x - 1)^15
detW[7] := x^35 (x^2 - x + 1) (x^4 + x^3 + x^2 + x + 1)^2 (x^2 + 1)^3 (x^2 + x + 1)^5 (x + 1)^9 (x - 1)^21
detW[8] := x^56 (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) (x^2 - x + 1)^2 (x^4 + x^3 + x^2 + x + 1)^3 (x^2 + 1)^4 (x^2 + x + 1)^7 (x + 1)^12 (x - 1)^28
detW[9] := x^84 (x^4 + 1) (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)^2 (x^2 - x + 1)^3 (x^4 + x^3 + x^2 + x + 1)^4 (x^2 + 1)^6 (x^2 + x + 1)^9 (x + 1)^16 (x - 1)^36

Nice...

Apparently, the fact that (x-1) (x^2 + x + 1) = x^3 - 1 is a hint (?).

Any ideas at all?
If it helps, W is an nxn matrix whose (i-j)th entry is x^((i-1)(j-1)).
2. Can not understand your work
3. I would google van der Monde determinant if I were you - as your W[n] is a special case of that determinant.
4. The vandermonde matrix has a particularly beautiful determinant. I'm not sure what relation your matrix has to a vandermonde matrix (if any; but I would trust RichE's judgment). Are you sure there is some general conjecture for the determinant of W[n]? I've used expansion by minors and there doesn't seem to be a discernable pattern.
5. Maybe if I was to rewrite the determinants as:

1
x-1
(x^2-x)(x^2-1)(x-1)
(x^3-x^2)(x^3-x)(x^3-1)(x^2-x)(x^2-1)(x-1)
...

it might become clearer.
6. (Original post by RichE)
Maybe if I was to rewrite the determinants as:

1
x-1
(x^2-x)(x^2-1)(x-1)
(x^3-x^2)(x^3-x)(x^3-1)(x^2-x)(x^2-1)(x-1)
...

it might become clearer.
Brilliant. What level math are you doing?

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: February 17, 2005
Today on TSR

Here's what to do next...

University open days

• University of East Anglia
All Departments Open 13:00-17:00. Find out more about our diverse range of subject areas and career progression in the Arts & Humanities, Social Sciences, Medicine & Health Sciences, and the Sciences. Postgraduate
Wed, 30 Jan '19
• Solent University
Sat, 2 Feb '19
• Sheffield Hallam University
Sun, 3 Feb '19
Poll
Useful resources

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

How to use LaTex

Writing equations the easy way

Study habits of A* students

Top tips from students who have already aced their exams

Chat with other maths applicants