If, S_i = aⁱ + bⁱ + cⁱ n what is value of \(\begin{vmatrix}S0 & S1 & S2 \\S1 & S2 & S3 \\S2 & S3 & S4 \end {vmatrix}\)?

We have, \(\begin{vmatrix}1 & 1 & 1 \\a & b & c \\a^2 & b^2 & c^2 \end {vmatrix}\)
So, the value of the \(\begin{vmatrix}1 & 1 & 1 \\a & b & c \\a^2 & b^2 & c^2 \end {vmatrix}\) = (a ? b)(b ? c)(c ? a)
Now, by circulant determinant,
\(\begin{vmatrix}1 & 1 & 1 \\a & b & c \\a^2 & b^2 & c^2 \end {vmatrix}\) X \(\begin{vmatrix}1 & 1 & 1 \\a & b & c \\a^2 & b^2 & c^2 \end {vmatrix}\) = \(\begin{vmatrix}S0 & S1 & S2 \\S1 & S2 & S3 \\S2 & S3 & S4 \end {vmatrix}\)
Multiplying the determinant in row by row,
We get, (a ? b)²(b ? c)²(c ? a)²