Ch7/MATH1804/YMC/200910
1
Chapter 7.
Matrices, determinants, system of linear equations,
eigenvalues and eigenvectors
7.1.
Matrix Arithmetic and Operations
This section is devoted to developing the arithmetic of matrices. We will see some of the differences
between arithmetic of real numbers and matrices.
Definition 7.1
Let
m
,
n
be positive integers. An
m
×
n
matrix
A
is an array of real numbers
A
=
a
11
a
12
· · ·
a
1
n
a
21
a
22
· · ·
a
2
n
.
.
.
.
.
.
.
.
.
.
.
.
a
m
1
a
m
2
· · ·
a
mn
where
a
ij
∈
R
is the
(
i, j
)
th entry of
A
.
We shall write
A
= (
a
ij
)
1
≤
i
≤
m
;1
≤
j
≤
n
for short, or
A
=
(
a
ij
)
m
×
n
, or
A
= (
a
ij
)
if the size of
A
is understood.
Matrices of the shape
m
×
1
are called
column vectors
, whereas matrices of the shape
1
×
n
are
called
row vectors
. Note that the integer
m
need not be equal to
n
. In the case of
m
=
n
, we have a
n
×
n
matrix, and it is called a
square matrix of order n
. The following are some examples of matrices:
A
=
a
11
a
12
a
13
a
14
a
21
a
22
a
23
a
24
!
,
B
=
1
9
2
7
5

3

5
8
4
,
C
=
4

6
5
1
Matrix
A
is a general
2
×
4
matrix with entries
a
ij
∈
R
where
1
≤
i
≤
2
and
1
≤
j
≤
4
. Matrix
B
is a
square matrix of order
3
, and
C
a
4
×
1
matrix (column vector).
Example 7.2
A
zero matrix
0
m
×
n
(or just
0
if the size is understood) is a matrix with all its entries
equal to zero, i.e.
a
ij
= 0
for
1
≤
i
≤
m
and
1
≤
j
≤
n
.
Example 7.3
A square matrix of order
n
is the
identity matrix
(of order
n
) if all the
diagonal entries
are one while all the
offdiagonal entries
are zero. We shall denote the identity matrix of order
n
by
I
n
, or simply
I
if there is no ambiguity. For example, when
n
= 2
,
3
, we have:
I
2
=
1
0
0
1
!
,
I
3
=
1
0
0
0
1
0
0
0
1
As we will see later, the zero and identity matrices play a similar role as
0
and
1
in the arithmetic
of real numbers.
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Ch7/MATH1804/YMC/200910
2
Definition 7.4
If
A
and
B
are both
m
×
n
matrices, then we say that
A
=
B
provided the corre
sponding entries from each matrix are equal, that is,
A
=
B
provided
a
ij
=
b
ij
for all
i
and
j
. Matrices
of different sizes cannot be equal.
Now we define addition and subtraction of matrices:
Definition 7.5
Let
A
= (
a
ij
)
m
×
n
and
B
= (
b
ij
)
m
×
n
. Then the
sum
and the
difference
of
A
and
B
,
written as
A
+
B
and
A

B
, are also
m
×
n
matrices with entries given by
a
ij
+
b
ij
and
a
ij

b
ij
respectively. Matrices of different sizes cannot be added or subtracted.
Example 7.6
1
3
7
2

2
3
!
+
0

1
5
2
3
1
!
=
1 + 0
3 + (

1)
7 + 5
2 + 2

2 + 3
3 + 1
!
=
1
2
12
4
1
4
!
.
Next we proceed to multiplication involving matrices. Note that we can define two kinds of mul
tiplication, namely scalar multiplication and matrix multiplication. We first look at scalar multiplication:
Definition 7.7
Let
A
= (
a
ij
)
m
×
n
.
For any
λ
∈
R
, the
scalar multiple
of
A
by
λ
is defined by
λA
= (
λa
ij
)
m
×
n
. In particular,
(

1)
A
is simply written as

A
.
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 Spring '10
 Prof
 Linear Algebra, Matrices, Row echelon form

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