设\(A\)是秩为\(2\)的\(3\)阶矩阵，\(\alpha\)是满足\(A\alpha = 0\)的非零向量，若对满足\(\beta^{T}\alpha = 0\)的\(3\)维列向量\(\beta\)，均有\(A\beta=\beta\)，则( )
A. \(A^{3}\)的迹为\(2\)
B. \(A^{3}\)的迹为\(5\)
C. \(A^{2}\)的迹为\(8\)
D. \(A^{2}\)的迹为\(9\)