Matrices,Matrix Non-Square Equations, Matrix Determinants, Linear Equations, Cramer's Rules, and a r

For a Science Fair I am trying to write a program that will a system (equation)

of 3 matrices of which only one is a square matrix:

[4 x 5] [5 x 4] = [4 x 4] Matrix

Inverses cannot be used. There is no inverse of non square matrices.

I may just settle for square matrices. Does anyone know a short algortihm

to calculate the determinant of a matrix? The inverse would be nice but

if it just the determinant that is fine.

If anyone know of Gauss[-Jordan] elimination algortihms for systems of equations

please say something. I would really appreciate.

Thanks,

Really Big Headache