Graphics Reference
In-Depth Information
−
the pattern
(fg
di)
is now very obvious. However, it is widely agreed that it is
best to ignore this and embrace a simple visual pattern. The price to be paid for this
is a negative sign as follows:
|
|=
−
−
−
+
−
A
a(ei
fh)
b(di
fg)
c(dh
eg).
4.9.2 The Laplace Expansion
The French mathematician, Pierre Simon de Laplace (1749-1827), developed a
method of expanding a determinant in terms of its minors, which, with the associ-
ated change of sign, is called a
cofactor
. The cofactor
c
row,col
of an element
a
row,col
is the minor that remains after removing from the original determinant the
row
row
and the
col
column.
For example, in (
4.17
) the minor of
a
11
is identified by removing the first row
and the first column; the minor of
a
12
is identified by removing the first row and the
second column; and the minor of
a
13
is identified by removing the first row and the
third column:
a
11
a
12
a
13
det
A
=
a
21
a
22
a
23
.
(4.17)
a
31
a
32
a
33
The three minor determinants for
a
11
,
a
12
and
a
13
are respectively:
a
22
a
23
a
21
a
23
a
21
a
22
A
11
=
,
12
=
,
13
=
a
32
a
33
a
31
a
33
a
31
a
32
whereas, the three cofactors are
c
11
=+
a
11
A
11
c
12
=−
a
12
A
12
c
13
=+
a
13
A
13
.
In general, the minor of
a
row,col
is denoted
A
row,col
.
Laplace proposed the following formulae for selecting the cofactor sign:
1
)
row
+
col
(
−
which generates the pattern
+−+
...
−+−
...
+−+
.
...
...
...
...
...
Although we have chosen the first row to expand the above determinants, any row,
or column may be used.
