Working with Arrays I Determinants

This chapter introduces the determinant as a mathematical object which we can use to tackle problems involving large numbers of simultaneous linear equations. By the end of this chapter, you should be able to  

We begin our discussion of linear systems by introducing the determinant as a tool for solving sets of simultaneous linear equations in which the indices of the unknown variables are all unity. Consider the pair of equations  

multiply equation (3.1) by 221 and equation (3.2) by a and subtract the resulting equations to yield  

The symbol on the left side of equation (3.7) defines a determinant of order 2, the expansion of which is given on the right. We can similarly express the numerators as determinants of order 2, and we see that the value of the two unknowns is then given by the ratio of two determinants  

The purpose of introducing this notation is that it readily extends to n linear algebraic equations in n unknowns. The problem then reduces to one of evaluating the respective determinants of order n. 

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