Example 1:

Suppose you want to encode and send the following message:

"STUDY LINEAR ALGEBRA"

First, using the conversion table, find the corresponding numbers associate with each letter.

STUDY LINEAR ALGEBRA

produces the following string of numbers:

19 20 21 4 25 12 9 14 5 1 18 1 12 7 5 2 18 1.

Then, divide the number into groups of 3 and write each group in the form of a $3\times 1$ vector.

$\left[ \begin{array}{r} 19\\ 20\\ 21\\ \end{array}\right]\hspace{.5cm} \lef... ...ight]\hspace{.5cm} \left[ \begin{array}{r} 2\\ 18\\ 1\\ \end{array}\right]$

The next step is to find the product of $A$ with any of these vectors:

$A \left[ \begin{array}{r} 19\\ 20\\ 21\\ \end{array}\right], A\left[ \begin{array}{r} 4\\ 25\\ 12\\ \end{array}\right], \cdots$.

Now you have the following vectors:

$\left[ \begin{array}{r} 19\\ 0\\ -18\\ \end{array}\right]\hspace{.5cm} \left... ...ht]\hspace{.5cm} \left[ \begin{array}{r} 35\\ -33\\ -19\\ \end{array}\right]$

This will provide you with the following string which can be sent.

19 0 -18 38 -34 -17 23 -14 -18 25 -34 -18 9 3 -14 35 -33 -19

The party who receives the message, should divide it into groups of 3 and form $3\times 1$ vectors and then multiplies each vector by $A^{-1}$.

For example $A^{-1}$ $\left[ \begin{array}{r} 19\\ 0\\ -18\\ \end{array}\right]=\left[ \begin{arra... ...\end{array}\right]=\left[ \begin{array}{r} 19\\ 20\\ 21\\ \end{array}\right]$

After obtaining a string of numbers, the conversion table can be used to convert the string to letters and obtain the decoded message.