Precalculus (6th Edition) Blitzer

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

Chapter 8 - Review Exercises - Page 951: 45

Answer

The decoded message is $18,\ 21\,12,\ 5$ or a RULE

Work Step by Step

First encode the given message as below: The word RULE is numerically equivalent to $18,21,12\text{ and 5}$. Now write these entries in the form of a square matrix as below: $\left[ \begin{matrix} 18 & 21 \\ 12 & 5 \\ \end{matrix} \right]$ Multiplying this matrix by the square matrix $ A=\left[ \begin{matrix} 3 & 2 \\ 4 & 3 \\ \end{matrix} \right]$ we get: $\begin{align} & \left[ \begin{matrix} 3 & 2 \\ 4 & 3 \\ \end{matrix} \right]\left[ \begin{matrix} 18 & 21 \\ 12 & 5 \\ \end{matrix} \right]=\left[ \begin{matrix} 3\left( 18 \right)+2\left( 21 \right) & 3\left( 12 \right)+2\left( 5 \right) \\ 4\left( 18 \right)+3\left( 21 \right) & 4\left( 12 \right)+3\left( 5 \right) \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 96 & 46 \\ 135 & 63 \\ \end{matrix} \right] \end{align}$ Now use these numbers, by columns, to write the encoded message $96,135,46,63$. Next decode this message by multiplying the multiplicative inverse of the coding matrix and the coded matrix as given below: It is given that the inverse of the coding matrix is, ${{A}^{-1}}=\left[ \begin{matrix} 3 & -2 \\ -4 & 3 \\ \end{matrix} \right]$
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