The transformation of coordinate from $(x, y, z)$ to $(x', y', z')$ is given by $ x' = a_{11}x + a_{12}y + a_{13}z\\ y' = a_{21}x + a_{22}y + a_{23}z\\ z' = a_{31}x + a_{32}y + a_{33}z$ If $(\hat i , \hat j, \hat k)$ and $(\hat i_1 , \hat j_1, \hat k_1)$ be unit vectors in two coordinates. Equating $ \vec r = \vec r'$ I got $\begin{bmatrix} \hat i\\ \hat j\\ \hat k \end{bmatrix} = \begin{bmatrix} a_{11} & a_{21} & a_{31}\\ a_{12} & a_{22} & a_{32}\\ a_{13} & a_{23} & a_{33} \end{bmatrix} \times \begin{bmatrix} \hat i_1\\ \hat j_1\\ \hat k_1 \end{bmatrix}$ The help manual says $\begin{bmatrix} \hat i_1\\ \hat j_1\\ \hat k_1 \end{bmatrix} = \begin{bmatrix} a_{11} & a_{21} & a_{31}\\ a_{12} & a_{22} & a_{32}\\ a_{13} & a_{23} & a_{33} \end{bmatrix} \times \begin{bmatrix} \hat i\\ \hat j\\ \hat k \end{bmatrix}$ I don't know if the manual is wrong or I am wrong. Please help.
The original problem is states as
Show that the operator $\nabla$ remains invariant passing from one rectangular Cartesian system of axes to another.