Cryptology is the science of developing secret codes and using those codes for encrypting and decrypting data. In June 1929, an article written by Lester S. Hill appeared in the American Mathematical Monthly. This was the first article that linked the fields of algebra and cryptology. Today, governments use sophisticated methods of coding and decoding messages. One type of code, which is extremely difficult to break, makes use of a large matrix to encode a message. The receiver of the message decodes it using the inverse of the matrix. This first matrix is called the encoding matrix and its inverse is called the decoding matrix. In this activity, we will use a simple method for encoding a message by first assigning a numeral to each letter of the alphabet. We will represent the letter A with the numeral 1 and continue to the letter Z which will be assigned the numeral 26. We will also assign the numeral 0 to a space in the message. For example, using the chart to the right, the word 1=9 = 0 A = 1 R = 18 S = 19 SYSTEM J = 10 can be written using numerals as K = 11 T = 20 B=2 C=3 19 25 19 20 5 13 L=12 U = 21 and then recorded in a matrix as D = 4 M = 13 V = 22 [19 25] E = 5 N = 14 W = 23 19 20. F = 6 O = 15 X = 24 5 13| G=7 P = 16 Y = 25 H=8 Q = 17 Z = 26 1. Example 1 (with word SYSTEM) To protect this message as it is transmitted, it is encoded by multiplying the message matrix by an encoding matrix, such as 4 3 2 a. [19 25] 19 20 [43] 2 = 5 13 b. Fill in the numerals for the new message: The receiver of this message can retrieve the original message by decoding it by using the inverse of the coding matrix. The inverse of a 2 X 2 matrix A = a b , provided det(A) #0, is A = 1 d -b] det(A) L-c a 2. What is the decoding matrix (the inverse of the encoding matrix)? b (Recall that det ([ 23). = ad-bc.) Fill in the information below. 61-288-88 =