Matrices are commonly used to encrypt data. Here is a simple form such an encryption can take. First, we represent each letter in the alphabet by a number, so let us take "space" = 0, A = 1, B = 2, and so on. Thus, for example, "ABORT MISSION" becomes [1 2 15 18 20 0 13 9 19 19 9 15 14].
To encrypt this coded phrase, we use an invertible matrix of any size with integer entries. For instance, let us take A to be the 2 x 2 matrix:
4 3
1 2
We can first arrange the coded sequence of numbers in the form of a matrix with two rows (using zero in the last place if we have an odd number of characters) and then multiply on the left by A.
Encrypted Matrix:
4 3
1 2
1 15
20 13
19 19
9 15
0 2
18 0
9 19
15 0
=
10 5 114 51 80 20 79 31 133 57 81 39 56 14
To decipher the encoded message, multiply the encrypted matrix by A^-1. The following question uses the above matrix A for encoding and decoding.
Decode the following message, which was encrypted using the matrix A. (Include any appropriate spaces in your answer.)
[57 33 126 54 29 11 80 20 46 29 145 65 74 41]