Example 2:

Consider an association of the alphabet with numbers in the following conversion table and given matrices $A$ and $A^{-1}$. Encode and decode the message " DO NOT COME"

$$\begin{array}{ccccccccccccc} A & B & C & D& E & F & G & H & I... ... & 7 & 9 &11 & 13 & 15 & 17 & 19 & 21 & 23 & 25 \\ \end{array}$$

$$\begin{array}{ccccccccccccc} N & O & P & Q & R & S & T & U & ... ... &29 &31& 33& 35& 37 &39 &41 &43 &45& 47 &49 &51\\ \end{array}$$

And let $A = \left[ \begin{array}{rr} 2&3\\ 3&5\\ \end{array}\right] , A^{-1}=\left[ \begin{array}{rr} 5&-3\\ -3&2\\ \end{array}\right]$.

a) Enoding the message:
First, using the coversion table above, find the corresponding string of numbers associate with this message.

PLEASE DO NOT COME

is equivalent to

31 23 9 1 37 9 7 29 27 29 39 5 29 25 9

Next, divide the string of numbers into pairs and group them in $2 \times 1$ vectors. Notice that the number of the letters in the plaintext is odd and is not divisible by 2, therefore, we add a dummy letter "E" to the end of the code and form eight different $2 \times 1$ vectors. The following string of numbers is produced by muliplying the eight $2 \times 1$ vectors by A.

131 208 21 32 101 156 101 166 141 226 93 142 133 212 45 72

and it is the message that should be transimited.

b) Decode the message:

Assuming that you have $A$ and $A^{-1}$, and the coversion table, and received the following message. You want to decode this message.

117 190 93 140 97 150 185 292 205 328

First, divide it into pairs and form five $2 \times 1$ vectors. Then, multiplies each of these vectors by $A^{-1}$, and form the following string of numbers:

15 29 45 1 35 9 49 29 41 41

Finally using the conversion table, find letters of alphebet corresponding to these numbers, as a result, you will get

H O W A R E Y O U U

which will read as

HOW ARE YOU.