The name of matrix operation If I have a matrix:
$$
A =
\begin{bmatrix} 
a_{11} &amp; a_{12} &amp; a_{13} \\
a_{21} &amp; a_{22} &amp; a_{23} \\
\end{bmatrix}
$$

and a vector:
$$
B =
\begin{bmatrix} 
b_1 \\
b_2 \\
\end{bmatrix}
$$

What is the name of the operation, which produces a vector:

$$
\begin{bmatrix} 
b_1 \cdot a_{11} + b_1 \cdot a_{12} + b_1 \cdot a_{13} \\
b_2 \cdot a_{21} + b_2 \cdot a_{22} + b_2 \cdot a_{23} \\
\end{bmatrix}
$$
Since your operation is invariant under row-permutations of matrix $A$ and only the sum matters then given $A$ we have
$$
\tilde{A} = \begin{pmatrix} a_{11} + a_{12} + a_{13} \\ a_{21} + a_{22} + a_{23} \end{pmatrix}
$$
and $b = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix}$. Now your operation becomes the Hadamard product of $\tilde{A}$ and $b$.
This is a dot product, also called Matrix Multiplication when it involves a vector and a matrix, or two matrices. 

In wikipedia article on Matrix Multiplication, in the section Definition, you have the following formula for the elements $c_{ij}$ of a matrix $\bf C=\bf A \bf B$ for $\bf A, \bf B$ and $\bf C$ all $m \times m$ matrices.

$$
c_{ij} = \sum_{k=1}^m a_{ik} b_{kj}
$$ 

If you treat your $\bf B$ matrix as a vector $\bf b$, with $m$ entries, instead of an $m \times m$ matrix, the operation will produce a vector $\bf c$ with $c_i$ elements given by:

$$
c_i = \sum_{k=1}^m a_{ik} b_k
$$

Which is exactly what you want.