Page 67 - Statistics and Data Analysis in Geology

Matrix Algebra

You will note that the denominators are the same for both unknowns. They also
are the determinants of the matrix A. That is,

Furthermore, the numerators can be expressed as determinants. For the equation
of XI, the numerator is the determinant of the matrix

and for x2, it is the determinant of

This procedure can be generalized to any set of simultaneous equations and
provides one common method for their solution. This procedure for solving equa-
tions is called Cramer’s rule. The rule states that the solution for any unknown xi
in a set of simultaneous equations is equal to the ratio of the two determinants.
The denominator is the determinant of the coefficients (in our example, the a’s).
The numerator is the same determinant except that the ith column is replaced by
the vector of right-hand terms (the vector of b’s). Let us check the rule with an

The denominators of the ratios for both unknown coefficients are the same:

1 1: i: 1 = (4 x 30) - (lox 10) = 20

The numerator of XI is the determinant

38
I110 301 lo (38 x 30) - (110 x 10) = 40
=
so x1 = 40/20 = 2. For XZ, the numerator is the determinant

38
I10 1101 = (4 x 110) - (10 x 38) = 60

so x2 = 60/20 = 3. These are the same unknowns we recovered by matrix inversion.
The determinant of an arbitrary square matrix such as the 3 x 3 example above
may be a positive value, a negative value, or zero. If the matrix is symmetric (the
variety of matrix we will encounter most often), its determinant cannot be negative.
However, the distinction between a positive determinant and a zero determinant is
very important because a matrix whose determinant is zero cannot be inverted by
ordinary methods. That is, the matrix will be singular.

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