I have coded recently the simplex algorithm. It is an algorithm that solves the optimizations of a problem of linear programming. It gives a function to minimize and constrains the variables with an inequality form.
It can be proven that the simplex algorithm solves this problem in Non-Polynomial time.
I have done a project for my math class where we analyze the linear systems. We have classified the first and second-order linear systems and give a general formula for an n-linear system. We have used algebra notation and we have encountered with the companion matrix power to the n problem.
The n-linear systems are used widely in finance for loans, interests, etc.