Solution
= Solution
Interchanging the rows and columns gives the <matrix transpose>
$$
\boxed{A^{\mathsf T}=\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}.}
$$
It cycles the three <basis vectors> in the opposite direction. Hence
$$
\boxed{A^{\mathsf T}\mathbf x=\begin{pmatrix}0\\0\\1\end{pmatrix},\quad
(A^{\mathsf T})^2\mathbf x=\begin{pmatrix}0\\1\\0\end{pmatrix},\quad
(A^{\mathsf T})^3\mathbf x=\begin{pmatrix}1\\0\\0\end{pmatrix}.}
$$