Chapter 8, Problem 142RE

a.

To determine

Find the given matrices for the given message.

Answer to Problem 142RE

The sequence of thematrices is $\left[\begin{array}{ccc}8& 5& 1\end{array}\right]$

  $\left[\begin{array}{ccc}4& 0& 4\end{array}\right]$

  $\left[\begin{array}{ccc}21& 5& 0\end{array}\right]$

  $\left[\begin{array}{ccc}23& 5& 19\end{array}\right]$

  $\left[\begin{array}{ccc}20& 0& 0\end{array}\right]$ .

Explanation of Solution

Given:

The given message is “HEADDUEWEST”.

Calculation:

Find the number for each letter.

  $\begin{array}{l}0\to \_14\to N\\ 1\to A15\to O\\ 2\to B16\to P\\ 3\to C17\to Q\\ 4\to D18\to R\\ 5\to E19\to S\\ 6\to F20\to T\\ 7\to G21\to U\\ 8\to H22\to V\\ 9\to I23\to W\\ 10\to J24\to X\\ 11\to K25\to Y\\ 12\to L26\to Z\\ 13\to M\end{array}$

Now

  $1\times 3$ matrices for the given message are

HEA

  $\to \left[\begin{array}{ccc}8& 5& 1\end{array}\right]$

D_ D

  $\to \left[\begin{array}{ccc}4& 0& 4\end{array}\right]$

UE_

  $\to \left[\begin{array}{ccc}21& 5& 0\end{array}\right]$

WES

  $\to \left[\begin{array}{ccc}23& 5& 19\end{array}\right]$

OW_

  $\to \left[\begin{array}{ccc}20& 0& 0\end{array}\right]$

Hence the sequence of thematrices is $\left[\begin{array}{ccc}8& 5& 1\end{array}\right]$

  $\left[\begin{array}{ccc}4& 0& 4\end{array}\right]$

  $\left[\begin{array}{ccc}21& 5& 0\end{array}\right]$

  $\left[\begin{array}{ccc}23& 5& 19\end{array}\right]$

  $\left[\begin{array}{ccc}20& 0& 0\end{array}\right]$ .

b.

To determine

Find the encodedmessage using encoding matrix.

Answer to Problem 142RE

The sequence of thecoded row matrices is $\left[\begin{array}{ccc}22& 47& 54\end{array}\right]$

  $\left[\begin{array}{ccc}0& -8& -20\end{array}\right]$

  $\left[\begin{array}{ccc}36& 77& 87\end{array}\right]$

  $\left[\begin{array}{ccc}19& 5& -42\end{array}\right]$

  $\left[\begin{array}{ccc}20& 40& 40\end{array}\right]$ .

Explanation of Solution

Given:

The given message is “HEAD DUE WEST” and the encoding matrix is $\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]$

Calculation:

Find the number for each letter.

  $\begin{array}{l}0\to \_14\to N\\ 1\to A15\to O\\ 2\to B16\to P\\ 3\to C17\to Q\\ 4\to D18\to R\\ 5\to E19\to S\\ 6\to F20\to T\\ 7\to G21\to U\\ 8\to H22\to V\\ 9\to I23\to W\\ 10\to J24\to X\\ 11\to K25\to Y\\ 12\to L26\to Z\\ 13\to M\end{array}$

Now

  $1\times 3$ matrices for the given message are

HEA

  $\to \left[\begin{array}{ccc}8& 5& 1\end{array}\right]$

Multiply by the given encoding matrix.

  $\begin{array}{l}\left[\begin{array}{ccc}8& 5& 1\end{array}\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]\\ =\left[\begin{array}{ccc}8\cdot 1+5\cdot 3+1\cdot \left(-1\right)& 8\cdot 2+5\cdot 7+1\cdot \left(-4\right)& 8\cdot 2+5\cdot 9+1\cdot \left(-7\right)\end{array}\right]\\ =\left[\begin{array}{ccc}22& 47& 54\end{array}\right]\end{array}$

D_ D

  $\to \left[\begin{array}{ccc}4& 0& 4\end{array}\right]$

Multiply by the given encoding matrix.

  $\begin{array}{l}\left[\begin{array}{ccc}4& 0& 4\end{array}\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]\\ =\left[\begin{array}{ccc}4\cdot 1+0\cdot 3+4\cdot \left(-1\right)& 4\cdot 2+0\cdot 7+4\cdot \left(-4\right)& 4\cdot 2+0\cdot 9+4\cdot \left(-7\right)\end{array}\right]\\ =\left[\begin{array}{ccc}0& -8& -20\end{array}\right]\end{array}$

UE_

  $\to \left[\begin{array}{ccc}21& 5& 0\end{array}\right]$

Multiply by the given encoding matrix.

  $\begin{array}{l}\left[\begin{array}{ccc}21& 5& 0\end{array}\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]\\ =\left[\begin{array}{ccc}21\cdot 1+5\cdot 3+0\cdot \left(-1\right)& 21\cdot 2+5\cdot 7+0\cdot \left(-4\right)& 21\cdot 2+5\cdot 9+0\cdot \left(-7\right)\end{array}\right]\\ =\left[\begin{array}{ccc}36& 77& 87\end{array}\right]\end{array}$

WES

  $\to \left[\begin{array}{ccc}23& 5& 19\end{array}\right]$

Multiply by the given encoding matrix.

  $\begin{array}{l}\left[\begin{array}{ccc}23& 5& 19\end{array}\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]\\ =\left[\begin{array}{ccc}23\cdot 1+5\cdot 3+19\cdot \left(-1\right)& 23\cdot 2+5\cdot 7+19\cdot \left(-4\right)& 23\cdot 2+5\cdot 9+19\cdot \left(-7\right)\end{array}\right]\\ =\left[\begin{array}{ccc}19& 5& -42\end{array}\right]\end{array}$

T_ _

  $\to \left[\begin{array}{ccc}20& 0& 0\end{array}\right]$

Multiply by the given encoding matrix.

  $\begin{array}{l}\left[\begin{array}{ccc}20& 0& 0\end{array}\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]\\ =\left[\begin{array}{ccc}20\cdot 1+0\cdot 3+0\cdot \left(-1\right)& 20\cdot 2+0\cdot 7+0\cdot \left(-4\right)& 20\cdot 2+0\cdot 9+0\cdot \left(-7\right)\end{array}\right]\\ =\left[\begin{array}{ccc}20& 40& 40\end{array}\right]\end{array}$

Hence the sequence of thecoded row matrices is $\left[\begin{array}{ccc}22& 47& 54\end{array}\right]$

  $\left[\begin{array}{ccc}0& -8& -20\end{array}\right]$

  $\left[\begin{array}{ccc}36& 77& 87\end{array}\right]$

  $\left[\begin{array}{ccc}19& 5& -42\end{array}\right]$

  $\left[\begin{array}{ccc}20& 40& 40\end{array}\right]$ .