Introduction

This chapter is about using matrices, not just building their algebra. The 19th-century mathematicians Carl Friedrich Gauss, Camille Jordan, Arthur Cayley and William Rowan Hamilton developed matrix theory precisely to investigate the solutions of systems of linear equations -- the workhorse computational problem behind circuit analysis, chemical-reaction balancing, economic input-output models, and countless other real situations.

Four solution methods for a system of linear equations are built up across the chapter, in this order:

1. Matrix inversion method -- $X=A^{-1}B$, needs $A$ square and non-singular.
2. Cramer's rule -- each unknown as a ratio of two determinants, needs $A$ square and non-singular.
3. Gaussian elimination method -- row-reduce the augmented matrix; works even when $A$ is singular or non-square.
4. Rank method -- the Rouché-Capelli consistency test $\rho(A)=\rho([A|B])$, the most general of all.

Before these can be developed, four supporting ideas must be in place: the inverse of a non-singular square matrix (via its adjoint), the rank of a matrix, elementary row and column transformations, and the consistency of a system of linear equations. The chapter builds each of these in turn, then uses them to solve non-homogeneous systems (§1.4, §1.5.1) and homogeneous systems (§1.5.2), the latter illustrated with a chemistry application: balancing a chemical reaction equation.