This quiz is about Vandermonde's determinants.
Define the two matrices (written as tables for convenience):
Wn(x1, x2, ... . xn) =
1 | 1 | 1 | ... | 1 |
x12 | x22 | x32 | ... | xn2 |
x13 | x23 | x33 | ... | xn3 |
... | ... | ... | ... | ... |
x1n | x2n | x3n | ... | xnn |
x1 + x | x2 + x | x3 + x | ... | xn + x |
x12 | x22 | x32 | ... | xn2 |
x13 | x23 | x33 | ... | xn3 |
... | ... | ... | ... | ... |
x1n | x2n | x3n | ... | xnn |
Prove that the determinants of these two matrices are
| An(x) | = | Vn | { 1 + ∑k=1 .. n (x/xn) }
These results can be generalized. Define the two matrices (again written as tables for convenience):
Wnk(x1, x2, ... . xn) =
1 | 1 | 1 | ... | 1 |
x1 | x2 | x3 | ... | xn |
x12 | x22 | x32 | ... | xn2 |
... | ... | ... | ... | ... |
x1k-1 | x2k-1 | x3k-1 | ... | xnk-1 |
x1k+1 | x2k+1 | x3k+1 | ... | xnk+1 |
... | ... | ... | ... | ... |
x1n | x2n | x3n | ... | xnn |
x1 + x | x2 + x | x3 + x | ... | xn + x |
x1 + x | x2 + x | x3 + x | ... | xn + x |
x12 + x | x22 + x | x32 + x | ... | xn2 + x |
... | ... | ... | ... | ... |
x1k + x | x2k + x | x3k + x | ... | xnk + x |
x1k+1 | x2k+1 | x3k+1 | ... | xnk+1 |
... | ... | ... | ... | ... |
x1n | x2n | x3n | ... | xnn |
Note:
Wn1 = Wn
An1(x) = An(x)
Ank(0) = Vn .
Prove that the determinants of these two matrices are (note that |Wn0 | = | Vn | )
| Ank(x) | = ∑ j=0 .. k xj |Wnj|
Marco Corvi - 2003