At this Wikipedia page it is stated under "dot product rule" that for $\mathbf{A,B}: \mathbb{R}^n \rightarrow \mathbb{R}^n $ smooth vector fields, it holds
$$ \nabla \langle \mathbf{A,B} \rangle =\mathbf{A \cdot J_B + B \cdot J_A} $$
where $\mathbf{J}$ is the jacobian matrix.
But on the righthand side of the equation I get "vector $\times$ squared matrix", which is not a valid row-column product. Is the formula formally wrong? Is the following formula correct?
$$ \nabla \langle \mathbf{A,B} \rangle = \mathbf{J_B \cdot A + J_A \cdot B} $$
Is there a reason they wrote it the other way around? (e.g.: further generalizations in manifolds...)?