If, x³ = 1, n, what will be value of\(\begin{vmatrix}a & b & c \\b & c & a \\c & a & b \end {vmatrix}\)?

We have, \(\begin{vmatrix}a & b & c \\b & c & a \\c & a & b \end {vmatrix}\)
As, x³ = 1,
= \(\begin{vmatrix}a & bx & cx^2 \\b & cx & ax^2 \\c & ax & bx^2 \end {vmatrix}\)
Replacing the 1^st column by C₁ + C₂ + C₃ we get,
= \(\begin{vmatrix}a + bx + cx^2 & bx & cx^2 \\ b + cx + ax^2 & cx & ax^2 \\c + ax + bx^2 & ax & bx^2 \end {vmatrix}\)
As, x³ = 1 so, x⁴ = x³ * x = x
= \(\begin{vmatrix}a + bx + cx^2 & b & c \\x^2 (a + bx + cx^2) & c & a \\x(a + bx + cx^2) & a & b \end {vmatrix}\)
= (a + bx + cx²)\(\begin{vmatrix}1 & b & c \\x^2 & c & a \\x & a & b \end {vmatrix}\)