Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 8 - Section 8.3 - Matrix Operations and Their Applications - Exercise Set - Page 918: 56

Answer

$\begin{align} \left[ \begin{matrix} 0 & \frac{3}{2} & \frac{3}{2} & \frac{1}{2} & \frac{1}{2} & 0 \\ 2 & 2 & \frac{5}{2} & \frac{5}{2} & \frac{9}{2} & \frac{9}{2} \\ \end{matrix} \right] \end{align}$ The graph is shown below:
1570416374

Work Step by Step

To reduce the perimeter of the graph above to half, we will multiply the matrix $B$ by $\frac{1}{2}$ as follows: $\begin{align} & \frac{1}{2}B=\frac{1}{2}\left[ \begin{matrix} 0 & 3 & 3 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 5 & 5 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 0 & \frac{3}{2} & \frac{3}{2} & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 0 & \frac{1}{2} & \frac{1}{2} & \frac{5}{2} & \frac{5}{2} \\ \end{matrix} \right] \end{align}$ To shift the reduced figure up by 2 units, we will add the following matrix, which represents the reduced figure. $\left[ \begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 2 & 2 & 2 & 2 & 2 \\ \end{matrix} \right]$ And the required coordinates to the matrix above and the resultant matrix will be: $\begin{align} & \left[ \begin{matrix} 0 & \frac{3}{2} & \frac{3}{2} & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 0 & \frac{1}{2} & \frac{1}{2} & \frac{5}{2} & \frac{5}{2} \\ \end{matrix} \right]+\left[ \begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 2 & 2 & 2 & 2 & 2 \\ \end{matrix} \right]=\left[ \begin{matrix} 0+0 & \frac{3}{2}+0 & \frac{3}{2}+0 & \frac{1}{2}+0 & \frac{1}{2}+0 & 0+0 \\ 0+2 & 0+2 & \frac{1}{2}+2 & \frac{1}{2}+2 & \frac{5}{2}+2 & \frac{5}{2}+2 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 0 & \frac{3}{2} & \frac{3}{2} & \frac{1}{2} & \frac{1}{2} & 0 \\ 2 & 2 & \frac{5}{2} & \frac{5}{2} & \frac{9}{2} & \frac{9}{2} \\ \end{matrix} \right] \end{align}$ The required coordinates to draw the shifted letter L are as follows: $\left( 0,2 \right),\left( \frac{3}{2},2 \right),\left( \frac{3}{2},\frac{5}{2} \right),\left( \frac{1}{2},\frac{5}{2} \right),\left( \frac{1}{2},\frac{9}{2} \right)$ and $\left( 0,\frac{9}{2} \right)$. Plot the points and trace them to obtain the curve. By subtracting the matrix $\left[ \begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 2 & 2 & 2 & 2 & 2 \\ \end{matrix} \right]$ from matrix $\frac{1}{2}B$, and plotting the obtained coordinates, the perimeter of the graph traced was reduced to half and shifted 2 units up from the original.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.