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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]](https://tex.z-dn.net/?f=%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7Da_%7B11%7D%20%26%20a_%7B12%7D%20%5C%5Ca_%7B21%7D%20%26%20a_%7B22%7D%20%5Cend%7Barray%7D%5Cright%5D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7Db_%7B11%7D%20%26%20b_%7B12%7D%20%5C%5Cb_%7B21%7D%20%26%20b_%7B22%7D%20%5Cend%7Barray%7D%5Cright%5D%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%20a_%7B11%7Db_%7B11%7D%20%2B%20a_%7B12%7Db_%7B21%7D%20%26%20a_%7B11%7Db_%7B12%7D%20%2B%20a_%7B12%7Db_%7B22%7D%20%5C%5C%20a_%7B21%7Db_%7B11%7D%20%2B%20a_%7B22%7Db_%7B21%7D%20%26%20a_%7B21%7Db_%7B12%7D%20%2B%20a_%7B22%7D%20b_%7B22%7D%20%5Cend%7Barray%7D%5Cright%5D%20)
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:
![\begin{array}{ccc} {} & [1] & [2] \\ 1| &1 & 2 \\ 2| & 2 &4 \end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bccc%7D%20%7B%7D%20%26%20%5B1%5D%20%26%20%5B2%5D%20%5C%5C%201%7C%20%261%20%26%202%20%5C%5C%202%7C%20%26%202%20%264%20%5Cend%7Barray%7D)
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]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7Da_%7B11%7D%20%26%20a_%7B12%7D%20%5C%5Ca_%7B21%7D%20%26%20a_%7B22%7D%20%5Cend%7Barray%7D%5Cright%5D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7Db_%7B11%7D%20%26%20b_%7B12%7D%20%5C%5Cb_%7B21%7D%20%26%20b_%7B22%7D%20%5Cend%7Barray%7D%5Cright%5D%20)
![= \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}](https://tex.z-dn.net/?f=%3D%20%5Cbegin%7Barray%7D%7Bcc%7D%20%7B%7D%20%26%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%20b_%7B11%7D%20%26%20b_%7B12%7D%5Cend%7Barray%7D%20%5Cright%5D%5C%5C%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%20a_%7B11%7D%20%5C%5C%20a_%7B21%7D%20%5Cend%7Barray%7D%20%5Cright%5D%20%5Cend%7Barray%7D%20%2B%5Cbegin%7Barray%7D%7Bcc%7D%20%7B%7D%20%26%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%20b_%7B21%7D%20%26%20b_%7B22%7D%5Cend%7Barray%7D%20%5Cright%5D%5C%5C%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%20a_%7B12%7D%20%5C%5C%20a_%7B22%7D%20%5Cend%7Barray%7D%20%5Cright%5D%20%5Cend%7Barray%7D)
![= \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]](https://tex.z-dn.net/?f=%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%20a_%7B11%7D%20b_%7B11%7D%20%26%20a_%7B11%7D%20b_%7B12%7D%20%5C%5C%20a_%7B21%7D%20b_%7B11%7D%20%26%20a_%7B21%7D%20b_%7B12%7D%20%5Cend%7Barray%7D%20%5Cright%5D%20%2B%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%20a_%7B12%7D%20b_%7B21%7D%20%26%20a_%7B12%7D%20b_%7B22%7D%20%5C%5C%20a_%7B22%7D%20b_%7B21%7D%20%26%20a_%7B22%7D%20b_%7B22%7D%20%5Cend%7Barray%7D%20%5Cright%5D)
Adding these matrices together, we get the exact same expression as the traditional definition.
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.