Systems of firstorder linear equations: I
liu dongwen
email: [email protected]
Before introducing the general theory of systems of ﬁrst order linear equations,
let us make a review of linear algebra.
Properties of matrices.
An
m
×
n
matrix
A
consists of a rectangular ar
ray of numbers, arranged in
m
rows and
n
columns, that is
A
=
a
11
a
12
···
a
1
n
a
21
a
22
···
a
2
n
.
.
.
.
.
.
.
.
.
a
m
1
a
m
2
···
a
mn
.
The notation (
a
ij
) is used to denote the matrix whose generic element is
a
ij
.
Let
A
= (
a
ij
)
,
B
= (
b
ij
)
,
C
= (
c
ij
) in the sequel.
1.
Transpose. A
T
= (
a
ji
)
.
2.
conjugate.
A
= (
a
ij
)
.
3.
Addition and subtraction.
If
A
and
B
are both
m
×
n
, then
A
±
B
= (
a
ij
±
b
ij
)
.
4.
Multiplication by a number.
Let
λ
be a scalar, then
λ
A
= (
λa
ij
)
.
5.
Multiplication.
If
A
is
m
×
n
and
B
is
n
×
r
, then
C
=
AB
is an
m
×
r
matrix, deﬁned by
c
ij
=
n
X
k
=1
a
ik
b
kj
.
In special, if
A
=
•
a b
c d
‚
,
x
=
•
x
1
x
2
‚
,
then
Ax
=
•
ax
1
+
bx
2
cx
1
+
dx
2
‚
.
In general, we have (
AB
)
T
=
B
T
A
T
,
AB
= (
A
)(
B
)
.
Note that in general
AB
6
=
BA
.
6.
Identity.
The multiplicative identity
I
, is given by
I
=
1 0
···
0
0 1
···
0
.
.
.
.
.
.
.
.
.
0 0
···
1
.
We have
AI
=
IA
=
A
for any (square) matrix
A
.
7.
Inverse and determinant.
A square matrix
A
is
invertible
if and only
if det
A
6
= 0
.
In this case there is another matrix
B
such that
AB
=
BA
=
I
.
This
B
is usually denoted by
A

1
.
For a 2
×
2 matrix
A
=
•
a b
c d
‚
,
1
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View Full Documentone has det
A
=
ad

bc.
In general, det
AB
= (det
A
)(det
B
)
.
If
A
is
n
×
n
and
λ
is a scalar, then det(
λ
A
) =
λ
n
det
A
.
Systems of linear algebraic equations.
A set of
n
simultaneous linear
algebraic equations in
n
variables,
a
11
x
1
+
a
12
x
2
+
···
+
a
1
n
x
n
=
b
1
,
.
.
.
a
n
1
x
1
+
a
n
2
x
2
+
···
+
a
nn
x
n
=
b
n
,
can be written as
Ax
=
b
,
(1)
where the
n
×
n
matrix
A
and the vector
b
are given, and the components of
x
are to be determined. The system is said to be
homogeneous
if
b
=
0
;
otherwise it is
nonhomogeneous
.
To solve a linear system like (1), we shall transform the augmented matrix
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