Overview

We now have the apparatus to consider how to compute the solution of linear equations in an arbitrary number of unknowns. We shall denote by the symbols Eq _i the following set of n equations in n unknowns x _i with constant real coefficients a _ij:

The given coefficients form a square matrix A = ( a _ij), and the unknowns x _i, and the given right-hand constants b _i are column vectors x and b, respectively. In matrix notation,

A very important quantity associated with a square matrix is its determinant. Presently, we shall see how to compute the determinant economically. For the time being we merely note that it may be defined recursively. Let A ^{( p,q)} be the matrix obtained by striking out the pth row and the qth column of A. Then,

for any column j; or alternatively,

for any row i.

This is known as Cramer's rule. It should not be used to compute the determinants of matrices of more than a few rows and columns: it requires n! multiplications to evaluate the determinant of an n n matrix, which rapidly becomes prohibitive.

If A and B are n n matrices, then det( AB) = det( A)det( B). Note also that if A ^T is the transpose of A, obtained...