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
.
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