The divergence of a vector field in Cartesian coordinate system (CCS) is defined as follows
$$ \mathrm{div}(\mathbf v) = \nabla \cdot \mathbf v = \begin{bmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{bmatrix} \cdot \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix} = \begin{bmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{bmatrix}^T \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix} = \begin{bmatrix} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix} =\frac{\partial v_1}{\partial x}+ \frac{\partial v_2}{\partial y}+ \frac{\partial v_3}{\partial z} $$
where $\cdot$ denotes the dot product; it was changed to transposition with matrix multiplication. Now lets define some matrix
$$A = \begin{bmatrix} a_{11} && a_{12} && a_{13} \\ a_{21} && a_{22} && a_{23} \\ a_{31} && a_{32} && a_{33} \\ \end{bmatrix} $$
How to calculate divergence of matrix in CCS and how looks its 'dot' product and 'matrix multiplication' form?
$$\mathrm{div}(A) = ?$$