There are many ways to encrypt a message. And the use of coding has become particularly significant in recent years (due to the explosion of the internet for example). One way to encrypt or code a message uses matrices and their inverse. Indeed, consider a fixed invertible matrix A. Convert the message into a matrix B such that AB is possible to perform. Send the message generated by AB. At the other end, they will need to know A-1 in order to decrypt or decode the message sent. Indeed, we have
which is the original message. Keep in mind that whenever an undesired intruder finds A, we must be able to change it. So we should have a mechanical way of generating simple matrices A which are invertible and have simple inverse matrices. Note that, in general, the inverse of a matrix involves fractions which are not easy to send in an electronic form. The best is to have both A and its inverse with integers as their entries. In fact, we can use our previous knowledge to generate such class of matrices. Indeed, if A is a matrix such that its determinant is
and all its entries are integers, then A-1 will have entries which are integers. So how do we generate such class of matrices? One practical way is to start with an upper triangular matrix with on the diagonal and integer-entries. Then we use the elementary row operations to change the matrix while keeping the determinant unchanged. Do not multiply rows with non-integers while doing elementary row operations.Info from: http://www.sosmath.com/matrix/coding/coding.html
catchy...this is a direct application of matrix into the real world...i like that
ReplyDelete