Chapter 8, Problem 15CT

To determine

The uncoded $1\times 3$ row matrices

Answer to Problem 15CT

The matrix notation to produce encoded message:

  $115-41-5914-3-1129-15-14128-53-604-40$

Explanation of Solution

Given information:

The message ‘KNOCK ON WOOD’ and encoding matrix below,

  $A=\left[\begin{array}{ccc}1& -1& 0\\ 1& 0& -1\\ 6& -2& -3\end{array}\right]$

Formula used:

  $\left[\text{Uncoded Matrix}\right]\left[\text{Encoding matrix}\right]=\left[\text{Coded Matrix}\right]$

Calculation:

Consider the message ‘KNOCK ON WOOD’ and encoding matrix below,

  $A=\left[\begin{array}{ccc}1& -1& 0\\ 1& 0& -1\\ 6& -2& -3\end{array}\right]$

Use 1 for A, 2 for B, 3 for C and so on. Make group of three alphabets to produce un-coded row matrices of order $1\times 3$ done below,

  $\begin{array}{l}\left[\begin{array}{ccc}11& 14& 15\end{array}\right]\left[\begin{array}{ccc}3& 11& 0\end{array}\right]\left[\begin{array}{ccc}15& 14& 0\end{array}\right]\left[\begin{array}{ccc}23& 15& 15\end{array}\right]\left[\begin{array}{ccc}4& 0& 0\end{array}\right]\\ KNOCKONWOOD\end{array}$

To encode a message multiply un-coded row matrices by encoding matrix A to obtain coded row matrices. That is,

  $\left[\text{Uncoded Matrix}\right]\left[\text{Encoding matrix}\right]=\left[\text{Coded Matrix}\right]$

So,

  $\begin{array}{l}\left[\begin{array}{ccc}11& 14& 15\end{array}\right]\left[\begin{array}{ccc}1& -1& 0\\ 1& 0& -1\\ 6& -2& -3\end{array}\right]=\left[\begin{array}{ccc}11+14+90& -11+0-30& 0-14-45\end{array}\right]\\ =\left[\begin{array}{ccc}115& -41& -59\end{array}\right]\end{array}$

  $\begin{array}{l}\left[\begin{array}{ccc}3& 11& 0\end{array}\right]\left[\begin{array}{ccc}1& -1& 0\\ 1& 0& -1\\ 6& -2& -3\end{array}\right]=\left[\begin{array}{ccc}3+11+0& -3+0+0& 0-11+0\end{array}\right]\\ =\left[\begin{array}{ccc}14& -3& -11\end{array}\right]\end{array}$

  $\begin{array}{l}\left[\begin{array}{ccc}15& 14& 0\end{array}\right]\left[\begin{array}{ccc}1& -1& 0\\ 1& 0& -1\\ 6& -2& -3\end{array}\right]=\left[\begin{array}{ccc}15+14+0& -15+0+0& 0-14+0\end{array}\right]\\ =\left[\begin{array}{ccc}29& -15& -14\end{array}\right]\end{array}$

  $\begin{array}{l}\left[\begin{array}{ccc}23& 15& 15\end{array}\right]\left[\begin{array}{ccc}1& -1& 0\\ 1& 0& -1\\ 6& -2& -3\end{array}\right]=\left[\begin{array}{ccc}23+15+90& -23+0-30& 0-15-45\end{array}\right]\\ =\left[\begin{array}{ccc}1528& -53& -60\end{array}\right]\end{array}$

  $\begin{array}{l}\left[\begin{array}{ccc}4& 0& 0\end{array}\right]\left[\begin{array}{ccc}1& -1& 0\\ 1& 0& -1\\ 6& -2& -3\end{array}\right]=\left[\begin{array}{ccc}4+0+0& -4+0+0& 0+0+0\end{array}\right]\\ =\left[\begin{array}{ccc}4& -4& 0\end{array}\right]\end{array}$

The sequence of coded row matrices is,

  $\left[\begin{array}{ccc}115& -41& -59\end{array}\right]\left[\begin{array}{ccc}14& -3& -11\end{array}\right]\left[\begin{array}{ccc}29& -15& -14\end{array}\right]\left[\begin{array}{ccc}128& -53& -60\end{array}\right]\left[\begin{array}{ccc}4& -4& 0\end{array}\right]$

Remove the matrix notation to produce encoded message:

  $115-41-5914-3-1129-15-14128-53-604-40$

Conclusion:

The matrix notation to produce encoded message:

  $115-41-5914-3-1129-15-14128-53-604-40$