Graphics Reference
In-Depth Information
4.9 Determinant of a Matrix
When solving a pair of simultaneous equations such as
+
=
ax
by
r
+
=
cx
dy
s
the expression
ad
bc
arises in the solution. For example, in the simultaneous
equations (
4.6
) and (
4.7
) the corresponding expression has a value 2
−
×
−
−
×
(
1
)
3
14 whose magnitude appears in the solution of
A
−
1
. Because this expression
is so useful, it is identified by the name
determinant
and is written
det
A
4
=−
=|
A
|=
ad
−
bc
where
ab
cd
.
A
=
Determinants are formed from square arrays, in that they have the same number
of rows and columns, which permits us to classify them in terms of their
order
.
Some texts classify a scalar quantity as a
first-order determinant
- for example
a
.
A
second-order determinant
has two rows and columns - for example
ab
cd
.
When dealing with three simultaneous equations
ax
+
by
+
cz
=
r
dx
+
ey
+
fz
=
s
gx
+
hy
+
iz
=
t
the corresponding matrix is
⎡
⎤
abc
def
gh i
⎣
⎦
A
=
and the equivalent determinant is
aei
+
bf g
+
cdh
−
ceg
−
af h
−
bdi.
4.9.1 Sarrus's Rule
The French mathematician, J.P. Sarrus (1789-1861), noted that a
third-order de-
terminant
is easily computed by exploiting a pattern which is very obvious if the
determinant's columns are extended as follows:
abcab
defde
gh i gh
.