How to multiply matrix with row vector?

How do I calculate a 3x3 matrix multiplication with a 3 column row vector, like below?

##
\begin{bmatrix}
A11 & A12 & A13\\
A21 & A22 & A23\\
A31 & A32 & A33
\end{bmatrix}\begin{bmatrix}
B1 & B2 & B3
\end{bmatrix}
##

You can only multiply matrix times column or row times matrix.
$$
\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}\cdot
\begin{pmatrix}b_{1}\\b_{2}\\b_{3}\end{pmatrix}=
\begin{pmatrix}a_{11}b_{1}+a_{12}b_{2}+a_{13}b_{3}\\a_{21}b_{1}+a_{22}b_{2}+a_{23}b_{3}\\a_{31}b_{1}+a_{32}b_{2}+a_{33}b_{3}\end{pmatrix}
$$

The "inner dimensions" must match. A column vector is a 3x1 [3 rows by 1 column] while a row vector is a 1x3 matrix.

Use the outer dimensions to get dimension of the resulting matrix.

So you can multiply a (1x3) by a (3x3) and get a 1x3.

Multiply a 1x3 by a 3x1 and get a 1x1... essentially the same as dot product of two vectors

I guess you can use various tricks, like assuming your row vector is the first row of a 3X3 matrix with zeros in the other places, or that the matrix on the left is made up of three “independent” column vectors, etc. However, you have to ask yourself what mathematical or physical objects do I get from such tricks?

apostolosdt said:

However, you have to ask yourself what mathematical or physical objects do I get from such tricks?

That's the point! It is important to understand the rule: 'row times column'. This has a meaning in itself, so "tricks" may cause more confusion than they solve. E.g.
$$
\begin{pmatrix}a&b&c\end{pmatrix}\cdot \begin{pmatrix}x\\y\\z\end{pmatrix} =\bigl \langle \begin{pmatrix}a&b&c\end{pmatrix}\, , \,\begin{pmatrix}x&y&z\end{pmatrix} \bigr\rangle = ax+by+cz \in \mathbb{R}
$$
is a number, e.g. a real number if the vectors have real components, whereas
$$
\begin{pmatrix}a\\b\\c\end{pmatrix}\cdot \begin{pmatrix}x&y&z\end{pmatrix}=\begin{pmatrix}a\\b\\c\end{pmatrix}\otimes \begin{pmatrix}x&y&z\end{pmatrix}=\begin{pmatrix}ax&ay&az\\bx&by&bz\\cx&cy&cz\end{pmatrix} \in \operatorname{M}(2,\mathbb{R})
$$
is a rank-##1## matrix, i.e. e.g. a ##(1,1)##-tensor.