Chapter 8.5, Problem 54E

(a)

To determine

To write: the uncoded $1\times 3$ row matrices for the massage.

Answer to Problem 54E

Uncoded matrices for the given massage are,

  $\left[935\right],\left[2518\right],\left[704\right],\left[514\right],\left[018\right],\left[514\right]$

Explanation of Solution

Given information:

Given massage

ICEBERG DEAD AHEAD

Given matrix

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

Calculation:

The message to be encoded is:

ICEBERG DEAD AHEAD

Partition the message including blank spaces into groups of three.

Thus, the uncoded row matrices will be,

  $\left[935\right],\left[2518\right],\left[704\right],\left[514\right],\left[018\right],\left[514\right]$

(b)

To determine

To encode: the massage using the encoding matrix.

Answer to Problem 54E

13 19 26 -1 -33 -77 3 -2 -14 4 1 -9 -5 -25 -47 4 1 -9

Explanation of Solution

Calculation:

The coded row matrices are obtained by multiplying each of the uncoded row matrices by the matrix $A$ , as follows:

Uncoded Matrix encoding matrix coded matrix

  $\begin{array}{c}\left[935\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]=\left[131926\right]\\ \left[2518\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]=\left[-1-33-77\right]\\ \left[704\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]=\left[3-2-14\right]\\ \left[514\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]=\left[41-9\right]\\ \left[51919\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]=\left[436748\right]\\ \left[018\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]=\left[-5-25-47\right]\\ \left[514\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]=\left[41-9\right]\end{array}$

So the sequence of coded row matrices

  $\left[131926\right],\left[-1-33-77\right],\left[3-2-14\right],\left[41-9\right],\left[-5-25-47\right],\left[41-9\right]$

Finally removing the matrix notation produces the following cryptogram,

13 19 26 -1 -33 -77 3 -2 -14 4 1 -9 -5 -25 -47 4 1 -9