Question

Suppose \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 + qx + r = 0 \) (with \( r \neq 0 \)) and they are in A.P. Then the rank of the matrix \( \begin{pmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{pmatrix} \) is:

• A. 3
• B. 2
• C. 0
• D. 1

Hint:

For matrix rank, use row reduction or determinant. For roots in A.P., express them symmetrically and check linear dependence in the matrix.