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

Feb 25, 2016

$768$

#### Explanation:

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

It is equal to
$D e t \left(V\right) = {\Pi}_{1 \le i \le j \le n} \left({\alpha}_{j} - {\alpha}_{i}\right)$, where $n$ is the number of rows (equal to the columns).

In the problem

$D e t \left(V\right) = \left({\alpha}_{4} - {\alpha}_{3}\right) \left({\alpha}_{4} - {\alpha}_{2}\right) \left({\alpha}_{4} - {\alpha}_{1}\right) \left({\alpha}_{3} - {\alpha}_{2}\right) \left({\alpha}_{3} - {\alpha}_{1}\right) \left({\alpha}_{2} - {\alpha}_{1}\right)$
$D e t \left(V\right) = \left(7 - 5\right) \left(7 - 3\right) \left(7 - 1\right) \left(5 - 3\right) \left(5 - 1\right) \left(3 - 1\right)$
$D e t \left(V\right) = 2 \cdot 4 \cdot 6 \cdot 2 \cdot 4 \cdot 2 = {2}^{3} \cdot {4}^{2} \cdot 6 = 8 \cdot 16 \cdot 6 = 768$