Given [mathbf{A}=left[begin{array}{ccc}0 & 1 & 0 0 & 0 & 1 -2 & -1 & -2end{array} ight],

Given

\[\mathbf{A}=\left[\begin{array}{ccc}0 & 1 & 0 \\0 & 0 & 1 \\-2 & -1 & -2\end{array}\right], \quad\mathbf{C}=\left[\begin{array}{lll}2 & 0 & \frac{1}{3}\end{array}\right]\]find the rank of\[\left[\begin{array}{lll}\mathbf{C}^{T} & \mathbf{A}^{T} \mathbf{C}^{T} &\left(\mathbf{A}^{T}\right)^{2} \mathbf{C}^{T}\end{array}\right]\]