The Chemistry Maths Book, Second Edition

584 Chapter 20Numerical methods

(2). The latter is therefore chosen as pivot equation. Similarly forx

2

in step 2, and

so on. A further refinement, called total pivoting, is to look for the largest relative

coefficient of any variable in each step.

Elimination method for the value of a determinant

A determinant can be reduced to triangular form by the first stage of the Gauss

elimination method. The value of the determinant is then the product of the diagonal

elements of the triangular form, as discussed in Section 17.6.

20.8 Gauss–Jordan elimination for the inverse of a matrix

The inverse matrix A

− 1

of a nonsingular matrix Aof order nsatisfies the matrix

equation

AA

− 1

1 = 1 I (20.47)

We write

A

− 1

1 = 1 (x

1

x

2

x

3

= x

n

), I 1 = 1 (e

1

e

2

e

3

= e

n

)

wherex

k

is the column vector formed from the elements of the kth column ofA

− 1

,

ande

k

is formed from the kth column of the unit matrix I. The matrix equation

(20.47) is then equivalent to the nmatrix equations

Ax

k

1 = 1 e

k

(k 1 = 1 1, 2, =,n) (20.48)

each of which represents a system of nsimultaneous linear equations, and each

of which can therefore be solved by Gauss elimination. The systematic way of doing

this is called Gauss–Jordan elimination, whereby an augmented matrix(A|I) is

transformed into (I|A

− 1

) by the two stage process demonstrated in the following

example.

9

EXAMPLE 20.17Gauss–Jordan elimination for the inverse of

The augmented matrix of the problem is

()AI| =−

−−

























123

212

311

100

010

001

A=−

−−

























123

212

311

9

Wilhelm Jordan (1842–1899), German geodesist.