One application of matrices is in the field of encryption and decryption. An encryption and decryption can be modeled using linear algebra methods by using an encrypting matrix A, a message x and an encrypted message b. Using this method allows for easy decryption of the message by multiplying b by the inverse of A.

This method of encryption is called a Hill cipher. In order to encrypt a message of n letters, each letter must be assigned to some numerical value and then multiplied by an nxn matrix A to create an n-length vector b with the encrypted numerical values.

Certain restrictions reside within our choice of the encryption matrix A. First, and obviously, A must be invertible, else once the message is encrypted, it will never be recovered again! Secondly, a method is needed to assign letters to numbers. A certain limited number of letters (the modulus) will be assigned to numbers. When we take the determinant of A, it must have no factors in common with the modulus, otherwise, again, the message cannot be decrypted.

As one might expect, this method of encryption isn’t especially powerful owing mainly to its linear nature. For example, if one truly wishes to discover the encryption matrix, s/he can simply input a known message and examine the output to discover facts about the encryption matrix (this is known as a plaintext attack). In short, knowing a small amount of both the encrypted text and unencrypted text can uncover the identity of the entire matrix.

So if you need a really cheap and easy way to encrypt something, go ahead and try the Hill Cipher. But don’t say that you haven’t been warned.

sources:
http://en.wikipedia.org/wiki/Matrix_encryption
http://en.wikipedia.org/wiki/Known-plaintext_attack