I am getting confused by two statements, namely $a\nabla$ and $\nabla\cdot c$ where
$$a = \left[\begin{array}{ccc}A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33}\end{array}\right]$$
$$c=\begin{bmatrix} A_{11} \\ A_{21} \\ A_{31} \end{bmatrix}$$
Is it $$ a\nabla= \left[\begin{array}{ccc}A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33}\end{array}\right]\begin{bmatrix}\partial_x \\ \partial_y \\ \partial_z \end{bmatrix} = \begin{bmatrix}A_{11}\partial_x + A_{12}\partial_y + A_{13}\partial_z \\ A_{21}\partial_x + A_{22}\partial_y + A_{23}\partial_z \\ A_{13}\partial_x + A_{23}\partial_y + A_{33}\partial_z \end{bmatrix} $$ ?
And if I have
$$\nabla\cdot c =\begin{bmatrix}\partial_x \\ \partial_y \\ \partial_z \end{bmatrix}\cdot\begin{bmatrix} A_{11} \\ A_{21} \\ A_{31} \end{bmatrix} = \partial_xA_{11} + \partial_y A_{21} + \partial_z A_{31} $$
Thanks in advance