Page 63 - Statistics and Data Analysis in Geology

Matrix Algebra

elements are the reciprocals of the corresponding elements of the original matrix.
Considering the 3 x 3 matrix, A,

-1
611 0 0 lla11 0

633 0 0 1/63
Certain combinations of otherwise complicated operations become very sim-
ple when the matrices involved are diagonal matrices. For example, consider the
multiplication
A-IA1/2 = A-112

If A is 3 x 3, the product is

In some applications, the inverse may not be required, but only the solutions
to a set of simultaneous equations. In the handworked example, we wanted the
values of the matrix X in the equation

To find this, we inverted A and then postmultiplied A-l by B to give X. We could have
instead found X directly by operating on B as A was transformed into an identity
matrix. To do this, we would utilize what is called an augmented matrix that has
one more column than it has rows. The column vector, B, then occupies the (n + 1)
column of the matrix, and the remaining (n n) part is inverted. Repeating the
x
same problem:

Matrices A and B are combined in an n x (n + 1) matrix.
381
10 30 110
1.0 2.5 Row one is divided by 4 and row two is divided by 10.
1.0 3.0 11.0

1.0 2.5 9.5 1 Row one is subtracted from row two.
0.0 0.5 1.5

Row two is multiplied by 5 and the product is subtracted
0.0 0.5 from row one.
5. [ Orno 1

0.0 1.0 Rowtwois dividedby0.5.

So, the (n + 1) column of the augmented matrix contains the solution to the si-
multaneous equation set, and our original matrix has been replaced by an identity
matrix.

135