Cryptography Cryptography is the use of mathematics to encode messages and prevent them from being read by anyone who doesn’t know the code. One way that this is accomplished is through the use of matrices.

Using Matrices as a Code Matrices can be used as a kind of secret code. The matrix is used to garble a message, and you can’t decode the garbled message without knowing the matrix that was used to garble it in the first place.

The Technique First, create a matrix that will be used to encode your message. You can use a square matrix of any size. We’ll use a 3 x 3 matrix. Second, determine the inverse of this matrix. This will be used to decode the message. Third, convert the message to a sequence of numbers. Assign each letter a number from 1-26, with spaces being 27. Group your numbers into 3 x 1 matrices (vectors). Multiply each of these vectors by your matrix to encode the message.

Example Let’s encode the message “Hello World” using the matrix

Example First, we need to calculate the inverse of this matrix. The process of calculating the inverse will be left up to the reader, but suffice it to say that the inverse is

Example Now we need to convert our message into numbers. If we let A = 1, B = 2, etc., we will end up with 8, 5, 12, 12, 15, 27, 23, 15, 18, 12, 4.

Example

Make sure you multiply in the correct order: Matrix * Vector, not Vector * Matrix.

Decoding To decode the matrix, just multiply each of the 3x1 matrices (vectors) that you received by the inverse matrix. This gives the original numbers. Try it yourself! Again, make sure you multiply in the correct order: Matrix * Vector.