Page 65 - Statistics and Data Analysis in Geology

Matrix Algebra

In a 2 x 2 matrix

we can find two combinations of elements that contain one and only one element
from each row and each column. These are a11a22 and a12a21.
The second subscripts in a11a22 are in correct numerical order and no rearrang-
ing is necessary. The number of transpositions is zero, so the sign of the product
is positive. However, a12a21 must be rearranged to a21a12 before the second sub-
scripts are in numerical order. This requires one transposition, so the product is
negative. The determinant of a 2 x 2 matrix is therefore

For a numerical example, we will consider the matrix
[: ;]

Next, let us consider a more complex example, a 3 x 3 determinant:

all 6.12 a13
a21 a22 6.23
a31 a32 a33
There are 3! , or 3 x 2 x 1 = 6, combinations of elements in a 3 x 3 matrix that
contain one element from each row and column and whose first subscripts are in
the order 1,2,3. Start with the top row and pick an entry from each row. Be sure to
choose in order from the first row, second row, third row, . . . nth row, with no more
than one entry from each column. All possible combinations that satisfy these
conditions in a 3 x 3 matrix are
all a22a33 all 623~2
a12a23a31 a12a21a33
a13a21a32 a13a~~a31
To determine the signs of each of these terms, we must see how many transposi-
tions are necessary to get the second subscripts in the order 1,2,3. For alla22a33,
no transpositions are necessary, so k = 0 and the term is positive, Transpositions
for the others and the resulting signs are given below:

all '2&2 ='llu32 u23 k= 1 sign=-
k=2 sign=+
u12 %&l ='la1 '23 ='31'12 '23
Ql@l a33 =% % a33 k= 1 sign=-
'1@1 '32 ='21 'la2 ='21'32 '13 k=2 sign=+
u13 u2a1 =ula1 uZ2 =a3, u1fi2 =u31 uZ2 u13 k = 3 sign = -

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