Page 66 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology- Chapter 3

Thus, there are three negative and three positive terms in the determinant. Sum-
ming according to the signs just found yields a single number, which is
-
+
+ ~~11~~2~33 alla~3m a12a23a31 - a12a21a33 + a13a21a32 - a13ma31
We can now try a matrix of real values:

4 3 2
2 4 1
1 0 3
The six terms possible are
(4 x 4 x 3) = 48
0
(4~1~0)=
(3xlxl)= 3
(3~2~3)=18
(2XZXO)= 0
(2X4X1)= 8
The first, third, and fifth of these require an even number of transpositions for
proper arrangement of the second subscript and so are positive. The others require
an odd number of transpositions and are therefore negative. Summing, we have

det A = +48 - 0 + 3 - 18 + 0 - 8 = 25

This method of evaluating a determinant is described by Pettofrezzo (1978). A
more conventional approach (see, for example, Anton and Rorres, 1994) uses what
is called the “method of cofactors,” but the two can be shown to be equivalent.
We now have at ow command a system for reducing a square matrix into its
determinant, but no clear grasp of what a determinant “really is.” Determinants
arise in many ways, but they appear most conspicuously during the solution of
sets of simultaneous equations. You may not have noticed them, however, because
they have been hidden in the inversion process we have been using.
Consider the set of equations:

a11x1+ al~x2 = bl
azm + mx2 = b2

Expressed in matrix form, this becomes

and we have discussed how the vector of unknown x’s can be solved by matrix
inversion. However, with algebraic rearrangement, the unknowns also can be found
by the equations
bla22 - alzb2
x1 =
a11a22 - a12a21
and

138