How do you marginalize in graphical model elimination?

Question

I'm reading Michael I. Jordan's book on probabilistic graphical models, and I don't understand the elimination algorithm presented in chapter 3. To narrow the question down, consider page 6. In equation (3.10), we see that $$m_5(x_2,x_3) = \sum_{x_5}p(x_5|x_3)p(\bar{x}_6|x_2,x_5)$$

where the $x_i$ are random variables and $\bar{x}_i$ indicates a fixed/realized value of $x_i$.

Given that all $x_i$ are discrete random variables (as is the case in chapter 3), both $p(x_5|x_3)$ and $p(\bar{x}_6|x_2,x_5)$ are represented by two-dimensional matrices. And since $m_5$ is a function of as-yet unrealized variables $x_2$ and $x_3$, it is also a two-dimensional matrix.

How then do we perform the multiplication and the summation above?