2 Multiplying two 2 by 2 matrices

If $A$ and $B$ are two matrices then the product $AB$ is obtained by multiplying the rows of $A$ with the columns of $B$ in the manner described above. This will only be possible if the number of elements in the rows of $A$ is the same as the number of elements in the columns of $B$ . In particular, we define the product of two $2\times 2$ matrices $A$ and $B$ to be another $2\times 2$ matrix $C$ whose elements are calculated according to the following pattern

$\left[\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right]\times \left[\begin{array}{cc}\hfill w\hfill & \hfill x\hfill \\ \hfill y\hfill & \hfill z\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill aw+by\hfill & \hfill ax+bz\hfill \\ \hfill cw+dy\hfill & \hfill cx+dz\hfill \end{array}\right]$

$AB=C$

The rule for calculating the elements of $C$ is described in the following Key Point:

We illustrate this construction for the abstract matrices $A$ and $B$ given above:

$\left[\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right]\times \left[\begin{array}{cc}\hfill w\hfill & \hfill x\hfill \\ \hfill y\hfill & \hfill z\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill \left[\begin{array}{c}\hfill ab\hfill \end{array}\right]\left[\begin{array}{c}\hfill w\hfill \\ \hfill y\hfill \end{array}\right]\hfill & \hfill \left[\begin{array}{c}\hfill ab\hfill \end{array}\right]\left[\begin{array}{c}\hfill x\hfill \\ \hfill z\hfill \end{array}\right]\hfill \\ \hfill \hfill \\ \hfill \left[\begin{array}{c}\hfill cd\hfill \end{array}\right]\left[\begin{array}{c}\hfill w\hfill \\ \hfill y\hfill \end{array}\right]\hfill & \hfill \left[\begin{array}{c}\hfill cd\hfill \end{array}\right]\left[\begin{array}{c}\hfill x\hfill \\ \hfill z\hfill \end{array}\right]\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill aw+by\hfill & \hfill ax+bz\hfill \\ \hfill cw+dy\hfill & \hfill cx+dz\hfill \end{array}\right]$

For example

$\left[\begin{array}{cc}\hfill 2\hfill & \hfill -1\hfill \\ \hfill 3\hfill & \hfill -2\hfill \end{array}\right]\times \left[\begin{array}{cc}\hfill 2\hfill & \hfill 4\hfill \\ \hfill 6\hfill & \hfill 1\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill \left[\begin{array}{c}\hfill 2-1\hfill \end{array}\right]\left[\begin{array}{c}\hfill 2\hfill \\ \hfill 6\hfill \end{array}\right]\hfill & \hfill \left[\begin{array}{c}\hfill 2-1\hfill \end{array}\right]\left[\begin{array}{c}\hfill 4\hfill \\ \hfill 1\hfill \end{array}\right]\hfill \\ \hfill \hfill \\ \hfill \left[\begin{array}{c}\hfill 3-2\hfill \end{array}\right]\left[\begin{array}{c}\hfill 2\hfill \\ \hfill 6\hfill \end{array}\right]\hfill & \hfill \left[\begin{array}{c}\hfill 3-2\hfill \end{array}\right]\left[\begin{array}{c}\hfill 4\hfill \\ \hfill 1\hfill \end{array}\right]\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill -2\hfill & \hfill 7\hfill \\ \hfill -6\hfill & \hfill 10\hfill \end{array}\right]$

Task!

Find the product $AB$ where $A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 4\hfill \end{array}\right]B=\left[\begin{array}{cc}\hfill 1& \hfill -1\\ \hfill -2& \hfill 1\end{array}\right]$

First write down row 1 of $A$ , column 2 of $B$ and form the first element in product $AB$ :

Answer

Answer

Finally find the two other elements of $C=AB$ and hence write down the matrix $C$ :

Answer

Clearly, matrix multiplication is tricky and not at all ‘natural’. However, it is a very important mathematical procedure with many engineering applications so must be mastered.