I separate these projects from those other using matrix operations, because I make a clear distinction between forward and inverse problems in my classes.
Cryptography - These projects come in two varieties: using modular arithmetic and not.
- Matrices with |determinant| = 1 - Uses matrix multiplication to encode a matrix and multiplication of the inverse to decode. Any matrix with determinant 1 or -1 will result in an inverse with integer components. Students tend to be drawn to these projects, but sometimes I find it hard to push them further, such as requiring them to create their own encoding matrices, etc. Resources abound.
- Modular arithmetic and row reduction - Uses matrix multiplication in modular arithmetic to encode and decode a message and row reduction in modular arithmetic to find the inverse. This is not that the decoded matrix is read using mod 26, but rather that the matrix operations are done with, say, mod 37. This project requires a little more tenacity on the part of the student, and this article by Keith Conrad discusses inverses of matrices under modular arithmetic.