2 The inverse of a 2 $\times$ 2 matrix

In this subsection we show how the inverse of a $2\times 2$ matrix can be obtained (if it exists).

Task!

Form the matrix products $AB$ and $BA$ where

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

Answer

You will see that had we chosen $C=\frac{1}{ad-bc}$ $\left[\begin{array}{cc}\hfill d& \hfill -b\\ \hfill -c& \hfill a\end{array}\right]$ instead of $B$ then both products $AC$ and $CA$ will be equal to $I$ . This requires $ad-bc\ne 0$ . Hence this matrix $C$ is the inverse of $A$ . However, note, that if $ad-bc=0$ then $A$ has no inverse . (Note that for the matrix $A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 0\hfill \end{array}\right],$ which occurred in the last task, $ad-bc=1\times 0-0\times 2=0$ confirming, as we found, that $A$ has no inverse.)

Task!

Which of the following matrices has an inverse?

$A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 3\hfill \end{array}\right],B=\left[\begin{array}{cc}\hfill 1& \hfill 1\\ \hfill -1& \hfill 1\end{array}\right],C=\left[\begin{array}{cc}\hfill 1& \hfill -1\\ \hfill -2& \hfill 2\end{array}\right],D=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

Answer

Task!

Find the inverses of the matrices $A,B$ and $D$ in the previous Task.

Use Key Point 8:

Answer

It can be shown that the matrix $A=\left[\begin{array}{cc}\hfill cos\theta & \hfill sin\theta \\ \hfill -sin\theta & \hfill cos\theta \end{array}\right]$ represents an anti-clockwise rotation through an angle $\theta$ in an $xy$ -plane about the origin. The matrix $B$ represents a rotation clockwise through an angle $\theta$ . It is given therefore by

$B=\left[\begin{array}{cc}\hfill cos\left(-\theta \right)& \hfill sin\left(-\theta \right)\\ \hfill -sin\left(-\theta \right)& \hfill cos\left(-\theta \right)\end{array}\right]=\left[\begin{array}{cc}\hfill cos\theta & \hfill -sin\theta \\ \hfill sin\theta & \hfill cos\theta \end{array}\right]$

Task!

Form the products $AB$ and $BA$ for these ‘rotation matrices’. Confirm that $B$ is the inverse matrix of $A$ .

Answer