0.2 Matrices  (Page 10/11)

We divide the letters of the message into groups of three.

ATT ACK –NO W––

Note that since the single letter "W" was left over on the end, we added two spaces to make it into a triplet.

Now we assign the numbers their corresponding letters from the table, and convert each triplet of numbers into $3\times 1$ matrices. We get

So far we have,

We multiply, on the left, each matrix of our message by the matrix B. For example,

By multiplying each of the matrices in ( III ) by the matrix B, we get the desired coded message as follows:

If we need to decode this message, we simply multiply the coded message by $B^{-1}$ , and associate the numbers with the corresponding letters of the alphabet.

Since this message was encoded by multiplying by the matrix B. We first determine inverse of B.

To decode the message, we multiply each matrix, on the left, by $B^{-1}$ . For example,

By multiplying each of the matrices in ( IV ) by the matrix $B^{-1}$ , we get the following.

Finally, by associating the numbers with their corresponding letters, we obtain the following.

And the message reads: HOLD FIRE.

The table below describes the above information.

In a matrix form it can be written as follows.

This matrix is called the input-output matrix . It is important that we read the matrix correctly. For example the entry $A_{\text{23}}$ , the entry in row 2 and column 3, represents the following.

$A_{\text{23}}=\text{20}\text{\%}$ of the tailor's production is used by the carpenter.

$A_{\text{33}}=\text{50}\text{\%}$ of the tailor's production is used by the tailor.