Unit 18 Determinants
Every square matrix has a number associated with it, called its determi
nant. In this section, we determine how to calculate this number, and also
look at some of the properties of the determinant of a (square) matrix.
Calculating the determinant of a small matrix is easy. But as the order
of the square matrix increases, the calculation of the determinant becomes
more complicated.
We deﬁne the determinant recursively. That is, for a square matrix of
order 1 or 2, we will deﬁne a simple formula for calculating the determinant.
However, for a square matrix of order 3 or higher, we deﬁne the determinant
of the matrix in terms of the determinants of certain submatrices. Thus, we
calculate the determinant of a large matrix by calculating the determinants
of various smaller matrices.
Deﬁnition 18.1.
The Deﬁnition of Determinant, Part 1 (
n
≤
2)
Let
A
=(
a
ij
)bean
n
×
n
matrix. Then A has associated with it a number,
called the determinant of
A
and denoted by
detA
.F
o
r
n
= 1 or 2, the value
of
detA
is calculated as follows:
If
n
=1
,then
detA
=
a
11
.
If
n
=2
detA
=
a
11
a
22

a
12
a
21
.
Example 1.
Find the determinants of the following matrices:
(a)
A
=[

4]
(b)
B
=
±
ab
cd
²
(c)
C
=
±
12

35
²
Solution:
(a) Here,
A
is 1
×
1, so
detA
=
a
11
=

4.
That is, for a 1
×
1 matrix, the determinant is the single number in the matrix.
1
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View Full Document(b) For a 2
×
2 matrix, we use the formula
detB
=
b
11
b
22

b
12
b
21
.Tha
ti
s
,w
e
take the product of the numbers going diagonally down to the right (i.e., on
the main diagonal) and then subtract from that the product of the numbers
going diagonally down to the left (i.e., on the other diagonal).
So assuming that
a
,
b
,
c
and
d
in matrix
B
are any scalars, the number
detB
is given by
detB
=
ad

bc
.
(c) Again, we have a 2
×
2 matrix, so we do the same calculation as in (b).
We get
detC
=
c
11
c
22

c
12
c
21
= (1)(5)

(2)(

3) = 5

(

6) = 11.
Now we are ready to deﬁne the notation and terminology needed to build
up to the deﬁnition of
detA
for
A
a square matrix of order
n
≥
3. We will
need to express certain
submatrices
of a matrix, as well as the determinant of
such a submatrix and also a particular scalar multiple of this number. These
are the focus of the next 2 deﬁnitions, after which we will be able to complete
our deﬁnition of determinant.
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 Fall '09
 OLDS,VICKY
 Calculus, Determinant, Characteristic polynomial, Diagonal matrix, Triangular matrix, detA

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