C H A P T E R
3
Determinants
Mathematics is the gate and key to the sciences.
— Roger Bacon
In this chapter, we introduce a basic tool in applied mathematics, namely the deter-
minant of a square matrix. The determinant is a number, associated with an
n
×
n
matrix
A
, whose value characterizes when the linear system
A
x
=
b
has a unique solution
(or, equivalently, when
A
−
1
exists). Determinants enjoy a wide range of applications,
including coordinate geometry and function theory.
Sections 3.1–3.3 give a detailed introduction to determinants, their properties, and
their applications. Alternatively, Section 3.4, “Summary of Determinants,” can provide a
nonrigorous and much more abbreviated introduction to the fundamental results required
in the remainder of the text. We will see in later chapters that determinants are invaluable
in the theory of eigenvalues and eigenvectors of a matrix, as well as in solution techniques
for linear systems of differential equations.
3.1
The Definition of the Determinant
We will give a criterion shortly (Theorem 3.2.4) for the invertibility of a square matrix
A
in terms of the determinant of
A
, written det
(A)
, which is a number determined directly
from the elements of
A
. This criterion will provide a first extension of the Invertible
Matrix Theorem introduced in Section 2.8.
To motivate the definition of the determinant of an
n
×
n
matrix
A
, we begin with
the special cases
n
=
1,
n
=
2, and
n
=
3.
Case 1:
n
=
1
.
According to Theorem 2.6.5, the 1
×
1 matrix
A
= [
a
11
]
is invertible
if and only if rank
(A)
=
1
,
if and only if the 1
×
1 determinant, det
(A)
, defined by
det
(A)
=
a
11
is nonzero.
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190
CHAPTER 3
Determinants
Case 2:
n
=
2
.
According to Theorem 2.6.5, the 2
×
2 matrix
A
=
a
11
a
12
a
21
a
22
is invertible if and only if rank
(A)
=
2
,
if and only if the row-echelon form of
A
has two
nonzero rows. Provided that
a
11
=
0, we can reduce
A
to row-echelon form as follows:
a
11
a
12
a
21
a
22
1
∼
a
11
a
12
0
a
22
−
a
12
a
21
a
11
.
1
. A
12
−
a
21
a
11
For
A
to be invertible, it is necessary that
a
22
−
a
12
a
21
a
11
=
0, or that
a
11
a
22
−
a
12
a
21
=
0.
Thus, for
A
to be invertible, it is necessary that the 2
×
2 determinant, det
(A)
, defined
by
det
(A)
=
a
11
a
22
−
a
12
a
21
(3.1.1)
be nonzero. We will see in the next section that this condition is also sufficient for the
2
×
2 matrix
A
to be invertible.
Case 3:
n
=
3
.
According to Theorem 2.6.5, the 3
×
3 matrix
A
=
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
is invertible if and only if rank
(A)
=
3, if and only if the row-echelon form of
A
has
three nonzero rows. Reducing
A
to row-echelon form as in Case 2, we find that it is
necessary that the 3
×
3 determinant defined by
det
(A)
=
a
11
a
22
a
33
+
a
12
a
23
a
31
+
a
13
a
21
a
32
−
a
11
a
23
a
32
−
a
12
a
21
a
33
−
a
13
a
22
a
31
(3.1.2)
be nonzero. Again, in the next section we will prove that this condition on det
(A)
is also
sufficient for the 3
×
3 matrix
A
to be invertible.

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