for n = 1, there's nothing to prove. so, from now on i'll assume that n > 1. we want to prove that your matrix has a non-zero

determinant. so suppose is the determinant of your matrix. first see that:

where

so we just need to prove that the proof is by induction on if then:

now suppose that whenever , and now define:

clearly is a polynomial of degree and

it's also evident that thus where is a constant.

so we only need to prove thatsince we only need to prove that this is easy to prove

because from the definition of it's easy to see that: but by

induction hypothesis thus