<s>[INST] Question about matrix multiplication notation I have the following matrices: 
$A=\begin{pmatrix}
        -\frac{2}{3} & \frac{1}{3} & 0 \\
        \frac{1}{6} & -\frac{1}{3} & \frac{1}{2} \\
        \frac{1}{6} & \frac{1}{3} & \frac{1}{2} \\
        \end{pmatrix}$
; $B=\begin{pmatrix}
        2 & 3\\
        2 & 0 \\
        0 & 3\\
        \end{pmatrix}$
; $C=\begin{pmatrix}
        6 & 3& 4 \\
        6 & 6 & 0 \\
        \end{pmatrix}$
;$u=\begin{pmatrix}2 \\ -5 \\ 4 \end{pmatrix}$
I want to calculate the following:
$\sum_j a_{ij}u_j$
$\sum_{jk} c_{ij}a_{jk}b_{kl}$
but I am not used to this kind of notation. Does the first one mean to multiply all $i$-rows of A with the $j$-column of u?
$\implies AB=\begin{pmatrix}
        -\frac{2}{3} & \frac{1}{3} & 0 \\
        \frac{1}{6} & -\frac{1}{3} & \frac{1}{2} \\
        \frac{1}{6} & \frac{1}{3} & \frac{1}{2} \\
        \end{pmatrix} \cdot \begin{pmatrix}2 \\ -5 \\ 4 \end{pmatrix}=\begin{pmatrix}-3 \\ 4 \\ 0 \end{pmatrix}$?
I am not sure what the second one is trying to say? Maybe $C\cdot A\cdot B$?
 [/INST] Yes. Note that you can perform the usual rows-columns product of two matrices $AB$ only if the number of  columns of $A$ is the same as the number of rows of $B$.
</s>