Question

How do you find the determinant of `((1, 1, 1, 1), (1, 3, 9, 27), (1, 5, 25, 125), (1, 7, 49, 343))`?

Answer 1

`768`

Explanation:

This special determinant is the so-called Vandermonde Determinant, described in this source:

Vandermonde Matrix and Determinant

It is equal to
`Det (V)=Pi_(1<=i<=j<=n) (alpha_j-alpha_i)`, where `n` is the number of rows (equal to the columns).

In the problem

`Det(V)=(alpha_4-alpha_3)(alpha_4-alpha_2)(alpha_4-alpha_1)(alpha_3-alpha_2)(alpha_3-alpha_1)(alpha_2-alpha_1)`
`Det(V)=(7-5)(7-3)(7-1)(5-3)(5-1)(3-1)`
`Det(V)=2*4*6*2*4*2=2^3*4^2*6=8*16*6=768`