Question

In matrix multiplication, such as AB=C, the columns of B form the rows of C. why is this?

Answer 1

Let's work with 2-by-2 matrices so we're on the same page. The ideas will work for any appropriate matrices.

From the rule of matrix multiplication, we see:
$\left[\begin{array}{cc}a_{11} & a_{12} \\a_{21} & a_{22} \end{array}\right] \left[\begin{array}{cc}b_{11} & b_{12} \\b_{21} & b_{22} \end{array}\right] = \left[\begin{array}{cc} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22} b_{22} \end{array}\right]$
As you noted, we see the columns of B contributing to the rows of C. The question is, why would we ever have defined matrix multiplication this way?

Here's a nontraditional way of feeling this connection. We can define matrix multiplication as "adding multiplication tables." A multiplication table is made by starting with a column and a row. For example,
$\begin{array}{ccc} {} & 1 & 2 \\ 1 & {} & {} \\ 2 & {} & {} \end{array}$
We then fill this table in by multiplying the row and column entries:
$\begin{array}{ccc} {} & [1] & [2] \\ 1| &1 & 2 \\ 2| & 2 &4 \end{array}$
It's then reasonable to say that given two matrices A and B, we can construct multiplication tables by taking the columns of A and pairing them with the rows of B:
$\left[\begin{array}{cc}a_{11} & a_{12} \\a_{21} & a_{22} \end{array}\right] \left[\begin{array}{cc}b_{11} & b_{12} \\b_{21} & b_{22} \end{array}\right]$

$= \begin{array}{cc} {} & \left[\begin{array}{cc} b_{11} & b_{12}\end{array} \right]\\ \left[\begin{array}{c} a_{11} \\ a_{21} \end{array} \right] \end{array} +\begin{array}{cc} {} & \left[\begin{array}{cc} b_{21} & b_{22}\end{array} \right]\\ \left[\begin{array}{c} a_{12} \\ a_{22} \end{array} \right] \end{array}$

$= \left[\begin{array}{cc} a_{11} b_{11} & a_{11} b_{12} \\ a_{21} b_{11} & a_{21} b_{12} \end{array} \right] + \left[\begin{array}{cc} a_{12} b_{21} & a_{12} b_{22} \\ a_{22} b_{21} & a_{22} b_{22} \end{array} \right]$

Adding these matrices together, we get the exact same expression as the traditional definition.