3 Some surprising results

We have already calculated the product $AB$ where

$A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 4\hfill \end{array}\right]\mathrm{\text{and}}B=\left[\begin{array}{cc}\hfill 1& \hfill -1\\ \hfill -2& \hfill 1\end{array}\right]$

Now complete the following task in which you are asked to determine the product $BA$ , i.e. with the matrices in reverse order.

Task!

For matrices $A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 4\hfill \end{array}\right]\mathrm{\text{and}}B=\left[\begin{array}{cc}\hfill 1& \hfill -1\\ \hfill -2& \hfill 1\end{array}\right]$ form the products of

row 1 of $B$ and column 1 of $A$ row 1 of $B$ and column 2 of $A$

row 2 of $B$ and column 1 of $A$ row 2 of $B$ and column 2 of $A$

Now write down the matrix $BA$ :

Answer

It is clear that $AB$ and $BA$ are not in general the same. In fact it is the exception that $AB=BA$ .

In the special case in which $AB=BA$ we say that the matrices $A$ and $B$ commute .

Task!

Calculate $AB$ and $BA$ where

$A=\left[\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right]\mathrm{\text{and}}B=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$

Answer

We call $B$ the $2\times 2$ zero matrix written 0 so that $A\times \underset{̲}{0}=\underset{̲}{0}\times A=0$ for any matrix $A$ .

Now in the multiplication of numbers, the equation

$ab=0$

implies that either $a$ is zero or $b$ is zero or both are zero. The following task shows that this is not necessarily true for matrices.

Task!

Carry out the multiplication $AB$ where

$A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right],B=\left[\begin{array}{cc}\hfill 1& \hfill -1\\ \hfill -1& \hfill 1\end{array}\right]$

Answer

Here we have a zero product yet neither $A$ nor $B$ is the zero matrix! Thus the statement $AB=0$ does not allow us to conclude that either $A=\underset{̲}{0}$ or $B=\underset{̲}{0}$ .

Task!

Find the product $AB$ where $A=\left[\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right]$ and $B=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

Answer

The matrix $\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ is called the identity matrix or unit matrix of order $2$ , and is usually denoted by the symbol $I.$ (Strictly we should write $I_2$ , to indicate the size.) $I$ plays the same role in matrix multiplication as the number 1 does in number multiplication.

Hence

just as $a\times 1=1\times a=a$ for any number $a$ , so $AI=IA=A$ for any matrix $A.$