Page 70 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 3

Let us assume that there are solutions to these equations other than the trivial
case where all the unknown x’s = 0. Look back at Cramer’s rule for the solution
of simultaneous equations, in which the unknowns are expressed as the ratio of
two determinants. Because the numerator in our present example would contain a
column of zeros, the determinant of the numerator also will be zero. That is, the
solution for the X vector is
0
x=-
IAl
Rewriting, this becomes
IAlX=O (3.7)

all - a12 a13
IA-hII= a21 a22-h a23 =O (3.8)
a3 1 a32 a33-

Thus we have

Because we know the various values of the elements aij, we can collect all of
these terms together in the form of an equation such as

where the (x’s represent the sum of the numerical values of the appropriate aij’s.
You should recognize that this is a quadratic equation of the general form

ax2 + bx + c = 0
which can be solved for the unknown terms by factoring. The general solution to a
quadratic equation is
-b+-
X= (3.10)
2a
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