You have already seen how Gaussian elimination turns a system of equations into an upper triangular form that is easy to solve by back substitution. In this chapter we take that idea a step further: we write the whole elimination process as a product of simple matrices. That product gives us a factorization of the original matrix—a kind of “factoring” into triangular pieces. This makes solving many systems with the same coefficient matrix much faster, and it shows a nice pattern, especially for symmetric matrices.