Chapter 7.8, Problem 29E

(a)

To determine

To find: The $1\times 2$ row matrices for the given message.

Answer to Problem 29E

The required $1\times 2$ row matrix are

  ${\left[\begin{array}{cc}3& 15\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}13& 5\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}8& 15\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}13& 5\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}19& 15\end{array}\right]}_{1\times 2}\text{and}{\left[\begin{array}{cc}15& 14\end{array}\right]}_{1\times 2}$ .

Explanation of Solution

Given information:

The given message is “COME HOME SOON”.

Consider the given message is COME HOME SOON.

Now, find the position of each of the alphabet.

  $\left[\begin{array}{cc}3& 15\\ C& O\end{array}\right]\left[\begin{array}{cc}13& 5\\ M& E\end{array}\right]\left[\begin{array}{cc}8& 15\\ H& O\end{array}\right]\left[\begin{array}{cc}13& 5\\ M& E\end{array}\right]\left[\begin{array}{cc}19& 15\\ S& O\end{array}\right]\left[\begin{array}{cc}15& 14\\ O& N\end{array}\right]$

Hence, the required $1\times 2$ row matrix are

  ${\left[\begin{array}{cc}3& 15\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}13& 5\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}8& 15\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}13& 5\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}19& 15\end{array}\right]}_{1\times 2}\text{and}{\left[\begin{array}{cc}15& 14\end{array}\right]}_{1\times 2}$ .

(b)

To determine

To find: The encoded matrix of the message.

Answer to Problem 29E

The required coded matrices are

  ${\left[\begin{array}{cc}48& 81\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}28& 51\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}53& 91\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}28& 51\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}64& 113\end{array}\right]}_{1\times 2}\text{and}{\left[\begin{array}{cc}47& 100\end{array}\right]}_{1\times 2}$

Explanation of Solution

Given information:

The given encodes matrix is

  $\left[\begin{array}{cc}1& 2\\ 3& 5\end{array}\right]$

Calculation:

From 29a. the $1\times 2$ row matrix are

  ${\left[\begin{array}{cc}3& 15\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}13& 5\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}8& 15\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}13& 5\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}19& 15\end{array}\right]}_{1\times 2}\text{and}{\left[\begin{array}{cc}15& 14\end{array}\right]}_{1\times 2}$ .

Now, multiply each of the $1\times 3$ row matrix with the given encoded matrix.

  $\left[\begin{array}{cc}1& 2\\ 3& 5\end{array}\right]$

  $\begin{array}{c}{\left[\begin{array}{cc}3& 15\end{array}\right]}_{1\times 2}{\left[\begin{array}{cc}1& 2\\ 3& 5\end{array}\right]}_{2\times 2}={\left[\begin{array}{cc}48& 81\end{array}\right]}_{1\times 2}\\ {\left[\begin{array}{cc}13& 5\end{array}\right]}_{1\times 2}{\left[\begin{array}{cc}1& 2\\ 3& 5\end{array}\right]}_{2\times 2}={\left[\begin{array}{cc}28& 51\end{array}\right]}_{1\times 2}\\ {\left[\begin{array}{cc}8& 15\end{array}\right]}_{1\times 2}{\left[\begin{array}{cc}1& 2\\ 3& 5\end{array}\right]}_{2\times 2}={\left[\begin{array}{cc}53& 91\end{array}\right]}_{1\times 2}\\ {\left[\begin{array}{cc}13& 5\end{array}\right]}_{1\times 2}{\left[\begin{array}{cc}1& 2\\ 3& 5\end{array}\right]}_{2\times 2}={\left[\begin{array}{cc}28& 51\end{array}\right]}_{1\times 2}\\ {\left[\begin{array}{cc}19& 15\end{array}\right]}_{1\times 2}{\left[\begin{array}{cc}1& 2\\ 3& 5\end{array}\right]}_{2\times 2}={\left[\begin{array}{cc}64& 113\end{array}\right]}_{1\times 2}\\ {\left[\begin{array}{cc}15& 14\end{array}\right]}_{1\times 2}{\left[\begin{array}{cc}1& 2\\ 3& 5\end{array}\right]}_{2\times 2}={\left[\begin{array}{cc}47& 100\end{array}\right]}_{1\times 2}\end{array}$

Hence, the required coded matrices are

  ${\left[\begin{array}{cc}48& 81\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}28& 51\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}53& 91\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}28& 51\end{array}\right]}_{1\times 2},{\left[\begin{array}{cc}64& 113\end{array}\right]}_{1\times 2}\text{and}{\left[\begin{array}{cc}47& 100\end{array}\right]}_{1\times 2}$ .