Chapter 2.4A, Problem 3GCE

To determine

To compare: the graph obtained by taking the product of matrices $CD$ with the given graph.

Answer to Problem 3GCE

The image is $270°$ counter clockwise rotation of given pre-image.

Explanation of Solution

Given information:

The graph of $AD$ matrix is given below.

  

The matrices given are as follows:

  $C=\left[\begin{array}{cc}\begin{array}{l}0\\ -1\end{array}& \begin{array}{l}\text{ 1}\\ 0\end{array}\end{array}\right]D=\left[\begin{array}{cccc}\begin{array}{l}1\\ -1\end{array}& \begin{array}{l}2\\ -2\end{array}& \begin{array}{l}3\\ -1\end{array}& \begin{array}{l}1\\ -1\end{array}\end{array}\right]$

Calculation:

Solve matrix $CD$ as follows:

  $\begin{array}{l}CD=\left[\begin{array}{cc}\begin{array}{l}0\\ -1\end{array}& \begin{array}{l}\text{ 1}\\ 0\end{array}\end{array}\right]\cdot \left[\begin{array}{cccc}\begin{array}{l}1\\ -1\end{array}& \begin{array}{l}2\\ -2\end{array}& \begin{array}{l}3\\ -1\end{array}& \begin{array}{l}1\\ -1\end{array}\end{array}\right]\\ =\left[\begin{array}{cccc}\begin{array}{l}-1\\ -1\end{array}& \begin{array}{l}-2\\ -2\end{array}& \begin{array}{l}-1\\ -3\end{array}& \begin{array}{c}-1\\ -1\end{array}\end{array}\right]\end{array}$

The coordinates of $CD$ are $\left(-1,-1\right),\left(-2,-2\right),\left(-1,-3\right)$ and $\left(-1,-1\right)$ . Plot the graph for the given coordinates as follows:

  

Comparing both the graph, we find that the image of plot 2 is $270°$ counter clockwise rotation of pre-image given in plot 1.