1 Multiplying row matrices and column matrices together

Let $A$ be a $1\times 2$ row matrix and $B$ be a $2\times 1$ column matrix:

$A=\left[\begin{array}{c}\hfill ab\hfill \end{array}\right]B=\left[\begin{array}{c}\hfill c\hfill \\ \hfill d\hfill \end{array}\right]$

The product of these two matrices is written $AB$ and is the $1\times 1$ matrix defined by:

$AB=\left[\begin{array}{c}\hfill ab\hfill \end{array}\right]\times \left[\begin{array}{c}\hfill c\hfill \\ \hfill d\hfill \end{array}\right]=\left[ac+bd\right]$

Note that corresponding elements are multiplied together and the results are then added together. For example

$\left[\begin{array}{c}\hfill 2-3\hfill \end{array}\right]\times \left[\begin{array}{c}\hfill 6\hfill \\ \hfill 5\hfill \end{array}\right]=\left[12-15\right]=\left[-3\right]$

This matrix product is easily generalised to other row and column matrices. For example if $C$ is a $1\times 4$ row matrix and $D$ is a $4\times 1$ column matrix:

$C=\left[\begin{array}{c}\hfill 2-432\hfill \end{array}\right]B=\left[\begin{array}{c}\hfill 3\hfill \\ \hfill 3\hfill \\ \hfill -2\hfill \\ \hfill 5\hfill \end{array}\right]$

then we define the product of $C$ with $D$ as

$CD=\left[\begin{array}{c}\hfill 2-432\hfill \end{array}\right]\times \left[\begin{array}{c}\hfill 3\hfill \\ \hfill 3\hfill \\ \hfill -2\hfill \\ \hfill 5\hfill \end{array}\right]=\left[6-12-6+10\right]=\left[-2\right]$

The only requirement is that the number of elements of the row matrix is the same as the number of elements of the column matrix.