Chapter 7, Problem 137CR

To determine

The uncoded row matrices for a given message and hence encode the message.

Answer to Problem 137CR

The uncoded row matrices are-

  $\left[\begin{array}{ccc}12& 15& 15\end{array}\right]$ $\left[\begin{array}{ccc}11& 0& 15\end{array}\right]$ $\left[\begin{array}{ccc}21& 20& 0\end{array}\right]$ $\left[\begin{array}{ccc}2& 5& 12\end{array}\right]$ $\left[\begin{array}{ccc}15& 23& 0\end{array}\right]$

The encoded message is given by the following array of numbers-

  $-21$ $6$ $0$ $-68$ $8$ $45$ $102$ $-42$ $-60$ $-53$ $20$ $21$ $99$ $-30$ $-69$

Explanation of Solution

Given: The message is LOOK OUT BELOW. And the given encoding matrix is-

  $A=\left[\begin{array}{ccc}2& -2& 0\\ 3& 0& -3\\ -6& 2& 3\end{array}\right]$ .

Concept Used: In cryptography, each letter starting from A to Z are assigned number 1-26 respectively and blank space is assigned to 0. As a result, any message can be partitioned and converted into an array of numbers from which groups of three numbers can be used to produce matrices of order $1\times 3$ which are called the uncoded matrices. The uncoded matrix is then multiplied to a given encoding matrix which gives the coded matrix. Finally all the numbers in the coded matrix can be written in an array forming a coded message.

Calculations: The given message is LOOK OUT BELOW. Partitioning the message (including blank spaces) into groups of three produces the following uncoded row matrices.

  $\left[\begin{array}{ccc}12& 15& 15\end{array}\right]$ $\left[\begin{array}{ccc}11& 0& 15\end{array}\right]$ $\left[\begin{array}{ccc}21& 20& 0\end{array}\right]$ $\left[\begin{array}{ccc}2& 5& 12\end{array}\right]$ $\left[\begin{array}{ccc}15& 23& 0\end{array}\right]$

L O O K O U T B E L O W

The encoding matrix is $A=\left[\begin{array}{ccc}2& -2& 0\\ 3& 0& -3\\ -6& 2& 3\end{array}\right]$ .

The coded matrices are obtained by multiplying each uncoded matrix with the encoding matrix. Hence, the required coded matrices are-

  $\left[\begin{array}{ccc}12& 15& 15\end{array}\right]\left[\begin{array}{ccc}2& -2& 0\\ 3& 0& -3\\ -6& 2& 3\end{array}\right]=\left[\begin{array}{ccc}-21& 6& 0\end{array}\right]$

  $\left[\begin{array}{ccc}11& 0& 15\end{array}\right]\left[\begin{array}{ccc}2& -2& 0\\ 3& 0& -3\\ -6& 2& 3\end{array}\right]=\left[\begin{array}{ccc}-68& 8& 45\end{array}\right]$

  $\left[\begin{array}{ccc}21& 20& 0\end{array}\right]\left[\begin{array}{ccc}2& -2& 0\\ 3& 0& -3\\ -6& 2& 3\end{array}\right]=\left[\begin{array}{ccc}102& -42& -60\end{array}\right]$

  $\left[\begin{array}{ccc}2& 5& 12\end{array}\right]\left[\begin{array}{ccc}2& -2& 0\\ 3& 0& -3\\ -6& 2& 3\end{array}\right]=\left[\begin{array}{ccc}-53& 20& 21\end{array}\right]$

  $\left[\begin{array}{ccc}15& 23& 0\end{array}\right]\left[\begin{array}{ccc}2& -2& 0\\ 3& 0& -3\\ -6& 2& 3\end{array}\right]=\left[\begin{array}{ccc}99& -30& -69\end{array}\right]$

Finally, removing the brackets and writing the coded matrices in an array gives the required coded message which is-

  $-21$ $6$ $0$ $-68$ $8$ $45$ $102$ $-42$ $-60$ $-53$ $20$ $21$ $99$ $-30$ $-69$