Real Life Application: Encoding a Message.

A cryptogram is a message written according to a secret code.  
The following technique uses matrices to encode and decode messages.
These can be fun to use in class when passing notes. This way, your
teacher can't read the message and you can argue that you were
just using your math, right?  
To begin, assign a number to each letter in the alphabet
(with 0 assigned to a space or break that represents the end of each word) 

i.e.:

0=_
1=A
2=B
3=C
4=D
5=E
6=F
7=G
8=H
9=I

10=J
11=K
12=L
13=M
14=N
15=O
16=P
17=Q
18=R

19=S
20=T
21=U
22=V
23=W
24=X
25=Y
26=Z

 


The massage is converted to numbers and partitioned into 1 x 2 uncoded 
row matrices.  For instance, the message "MATRICES" would be converted 
into the following row matrices:

[13 1]  [20 18]  [9  3]  [5 19]

To encode the message choose a 2 x 2 matrix (A) that has an inverse and 
multiply the uncoded row matrices (on the right) by A to obtain 
coded row matrices.

Ex. 1 Use the matrix 	to encode the massage MATRICES

Solution:

The coded row matrices are obtained by multiplying each of the uncoded row 
matrices (listed above) by the matrix A

 

Uncoded Row Matrix

[13 1]

[20 18]

[9 3]

[5 19]

Encoding Matrix A

Coded Matrix Row

[-11  23]

[16  -14]

[-3  9]

[33  -47] 

The message then becomes:  -11, 23, 16, -14, -3, 9, 33, -47




Decoding a Message

Obviously, if you are going to encode the message, you should know
how to decode it. It just makes life simpler. 
Use the inverse of Matrix A to decode the message

Solution:

To decode the message, partition the message into groups to 2 
numbers to form coded row matrices.  
Then multiply each coded row matrix by the inverse of A 
(on the right) to obtain the decoded row matrices.

Coded Row Matrix

[-11  23]

[16  -14]

[-3  9]

[33  -47]

Decoding inverse of A

Decoded Row Matrix

[13 1]

[20 18]

[9 3]

[5 19]

You try:


1)	Encode: BREAK A LEG

2)	Encode: GET WELL SOON

3)	Decode: 55, 19, 85, 33, 42, 14, 40, 15, 56, 20, 82, 29






Answers:

1)	-16, 50, 4, -7, 11, -22, 1, -2, 7, -9, 7, -14
2)	19, 31, 40, 60, 51, 79, 36, 60, 19, 38, 45, 75, 28, 42
3)	BONE NEEDLES

*Cramer's rule section is still under construction
and will not be posted at this time*