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.