5 Multiplying non-square matrices together

So far, we have just looked at multiplying $2\times 2$ matrices and $3\times 3$ matrices. However, products between non-square matrices may be possible.

Example 4

Find the product $AB$ if $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 2\hfill \\ \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill \end{array}\right]$ and $B=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 5\hfill \\ \hfill 6\hfill & \hfill 1\hfill \\ \hfill 4\hfill & \hfill 3\hfill \end{array}\right]$

Solution

Since $A$ is a $2\times 3$ and $B$ is a $3\times 2$ matrix the product $AB$ can be found and results in a $2\times 2$ matrix.

$AB=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 2\hfill \\ \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill \end{array}\right]\times \left[\begin{array}{cc}\hfill 2\hfill & \hfill 5\hfill \\ \hfill 6\hfill & \hfill 1\hfill \\ \hfill 4\hfill & \hfill 3\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill \left[\begin{array}{c}\hfill 122\hfill \end{array}\right]\left[\begin{array}{c}\hfill 2\hfill \\ \hfill 6\hfill \\ \hfill 4\hfill \end{array}\right]\hfill & \hfill \left[\begin{array}{c}\hfill 122\hfill \end{array}\right]\left[\begin{array}{c}\hfill 5\hfill \\ \hfill 1\hfill \\ \hfill 3\hfill \end{array}\right]\hfill \\ \hfill \hfill \\ \hfill \left[\begin{array}{c}\hfill 234\hfill \end{array}\right]\left[\begin{array}{c}\hfill 2\hfill \\ \hfill 6\hfill \\ \hfill 4\hfill \end{array}\right]\hfill & \hfill \left[\begin{array}{c}\hfill 234\hfill \end{array}\right]\left[\begin{array}{c}\hfill 5\hfill \\ \hfill 1\hfill \\ \hfill 3\hfill \end{array}\right]\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 22\hfill & \hfill 13\hfill \\ \hfill 38\hfill & \hfill 25\hfill \end{array}\right]$

Task!

Obtain the product $AB$ if $A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill -2\hfill \\ \hfill 2\hfill & \hfill -3\hfill \end{array}\right]$ and $B=\left[\begin{array}{ccc}\hfill 2\hfill & \hfill 4\hfill & \hfill 1\hfill \\ \hfill 6\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$

Answer