In this chapter we first define the Coates digraph of a square matrix. The Coates digraph is a slight variation of the digraph used in the previous chapter. We use the Coates digraph to give a nontraditional definition of the determinant of a square matrix. Using this definition, we derive the basic properties of a determinant that are useful in its evaluation. In particular, it is shown how the calculation of a determinant can be reduced to the calculation of determinants of lower order. We also derive the formula for the determinant that is used in its classical definition and actually establish the equivalence of the two definitions of the determinant. The determinant can be defined yet again in a third way-using the Ko¨nig digraph-a fact that will be useful later in the book. A special determinantal formula, derived in Section 4.3, will be used in Chapter 7. Section 4.5 describes the Laplace development of a determinant.