Solving linear equations

In general, solving a system of linear equations $$Ax=b$$ is pretty difficult. However, there are certain subsets of systems that are easy to solve. The idea is to take any system and convert it to this form that's easy to solve. Below are the main sets of systems that are easy to solve (we consider square matrices, $$A \in \mathbb R^{n\times n}$$):
 * $$A$$ is diagonal ($$x_i = b_i/a_{ii}$$)
 * $$A$$ is lower triangular (start with $$x_1 = b_1/a_{11}$$ and substitute it going forward)
 * $$A$$ is upper triangular (start with $$x_n = b_n/a_{nn}$$ and substitute it going backward)
 * $$A$$ is orthogonal ($$x=A^Tb$$)
 * $$A$$ is permutation matrix