Suppose \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 + qx + r = 0 \) (with \( r \ne 0 \)) 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:
Show Hint
For rank problems with roots:
\begin{itemize}
\item Use symmetric root conditions.
\item A.P. roots simplify nicely.
\item Check row sums for dependence.
\end{itemize}