Chapter 7.8, Problem 32E

(a)

To determine

To find The $1\times 3$ row matrices for the given message.

Answer to Problem 32E

The required $1\times 3$ row matrix are

  ${\left[\begin{array}{ccc}8& 1& 16\end{array}\right]}_{1\times 3},{\left[\begin{array}{ccc}16& 25& 2\end{array}\right]}_{1\times 3},{\left[\begin{array}{ccc}9& 18& 20\end{array}\right]}_{1\times 3},{\left[\begin{array}{ccc}8& 4& 1\end{array}\right]}_{1\times 3}\text{and}{\left[\begin{array}{ccc}25& 0& 0\end{array}\right]}_{1\times 3}$ .

Explanation of Solution

Given information:

The given message is “HAPPY BIRTHDAY”.

Consider the given message is HAPPY BIRTHDAY.

Now, find the position of each of the alphabet.

  $\left[\begin{array}{ccc}\begin{array}{l}8\\ H\end{array}& \begin{array}{l}1\\ A\end{array}& \begin{array}{l}16\\ P\end{array}\end{array}\right]\left[\begin{array}{ccc}\begin{array}{l}16\\ P\end{array}& \begin{array}{l}25\\ Y\end{array}& \begin{array}{l}2\\ B\end{array}\end{array}\right]\left[\begin{array}{ccc}\begin{array}{l}9\\ I\end{array}& \begin{array}{l}18\\ R\end{array}& \begin{array}{l}20\\ T\end{array}\end{array}\right]\left[\begin{array}{ccc}\begin{array}{l}8\\ H\end{array}& \begin{array}{l}4\\ D\end{array}& \begin{array}{l}1\\ A\end{array}\end{array}\right]\left[\begin{array}{ccc}\begin{array}{l}25\\ Y\end{array}& \begin{array}{l}0\\ 0\end{array}& \begin{array}{l}0\\ 0\end{array}\end{array}\right]$

Hence, the required $1\times 3$ row matrix are

  ${\left[\begin{array}{ccc}8& 1& 16\end{array}\right]}_{1\times 3},{\left[\begin{array}{ccc}16& 25& 2\end{array}\right]}_{1\times 3},{\left[\begin{array}{ccc}9& 18& 20\end{array}\right]}_{1\times 3},{\left[\begin{array}{ccc}8& 4& 1\end{array}\right]}_{1\times 3}\text{and}{\left[\begin{array}{ccc}25& 0& 0\end{array}\right]}_{1\times 3}$ .

(b)

To determine

To calculate: The encoded matrix of the message.

Answer to Problem 32E

The required coded matrices are

  ${\left[\begin{array}{ccc}-5& 87& -87\end{array}\right]}_{1\times 3},{\left[\begin{array}{ccc}-89& 215& 243\end{array}\right]}_{1\times 3},{\left[\begin{array}{ccc}43& 224& 40\end{array}\right]}_{1\times 3},{\left[\begin{array}{ccc}19& 64& 45\end{array}\right]}_{1\times 3}\text{and}{\left[\begin{array}{ccc}25& 50& 50\end{array}\right]}_{1\times 3}$

Explanation of Solution

Given information:

The given encodes matrix is

  $\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& 4& -7\end{array}\right]$

Calculation:

From 32a. the $1\times 3$ row matrix are

  ${\left[\begin{array}{ccc}8& 1& 16\end{array}\right]}_{1\times 3},{\left[\begin{array}{ccc}16& 25& 2\end{array}\right]}_{1\times 3},{\left[\begin{array}{ccc}9& 18& 20\end{array}\right]}_{1\times 3},{\left[\begin{array}{ccc}8& 4& 1\end{array}\right]}_{1\times 3}\text{and}{\left[\begin{array}{ccc}25& 0& 0\end{array}\right]}_{1\times 3}$ .

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

  $\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& 4& -7\end{array}\right]$

Hence, the required coded matrices are

  ${\left[\begin{array}{ccc}-5& 87& -87\end{array}\right]}_{1\times 3},{\left[\begin{array}{ccc}-89& 215& 243\end{array}\right]}_{1\times 3},{\left[\begin{array}{ccc}43& 224& 40\end{array}\right]}_{1\times 3},{\left[\begin{array}{ccc}19& 64& 45\end{array}\right]}_{1\times 3}\text{and}{\left[\begin{array}{ccc}25& 50& 50\end{array}\right]}_{1\times 3}$