In matrix multiplication, such as ab=c, the columns of b form the rows of c. why is this?

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:

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,

We then fill this table in by multiplying the row and column entries:

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:






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