If p , q , r are the cube roots of unity, then what is vmatrix p^2+q^2 & r^2 & r^2 p^2 & q^2+r^2 & p^2 q^2 & q^2 & r^2+p^2 vmatrix equal to ?

Step-by-step solution

Let a = p^2 , b = q^2 , and c = r^2 . The given determinant can be written as: = vmatrix a+b & c & c a & b+c & a b & b & c+a vmatrix Applying the row operation R₁ R₁ - R₂ - R₃ , we get: = vmatrix 0 & -2b & -2a a & b+c & a b & b & c+a vmatrix Expanding the determinant along the first row: = 0 - (-2b) vmatrix a & a b & c+a vmatrix + (-2a) vmatrix a & b+c b & b vmatrix = 2b(ac + a^2 - ab) - 2a(ab - b^2 - bc) = 2abc + 2a^2b - 2ab^2 - 2a^2b + 2ab^2 + 2abc = 4abc Substituting back a = p^2 , b = q^2 , and c = r^2 : = 4(p^