Chapter 1.
Matrices, determinants, system of linear equa
tions
1.1.
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 1.1
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
)
if the size of
A
is understood. Denote by
M
m
×
n
(
R
)
the set of all
m
×
n
matrices
with real entries.
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
. Denote by
M
n
(
R
)
the set
of all square matrices of order
n
with real entries. 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
is a
4
×
1
matrix (column vector).
Example 1.2
A
zero matrix
0
m
×
n
∈
M
m
×
n
(
R
)
(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
. Another special
matrix is the
n
×
n
identity matrix
whose entries
δ
ij
are the
Kronecker delta
:
δ
ij
=
1
if
i
=
j
;
0
if
i
6
=
j,
for
1
≤
i, j
≤
n
, i.e.
the
diagonal entries
are all one while the
offdiagonal entries
are all
zero. Often the identity matrices are denoted by
I
n
. As you will see later, the zero and identity
matrices play a similar role as 0 and 1 in the arithmetic of real numbers.
2
Definition 1.3
If
A, B
∈
M
m
×
n
(
R
)
then we say that
A
=
B
provided corresponding 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 1.4
Let
A, B
∈
M
m
×
n
(
R
)
, with
A
= (
a
ij
)
and
B
= (
b
ij
)
. 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.
Next we proceed to multiplication involving matrices. Note that we can define two kinds of
multiplication, namely scalar multiplication and matrix multiplication.
We first look at scalar
multiplication:
Definition 1.5
Let
A
= (
a
ij
)
m
×
n
∈
M
m
×
n
(
R
)
. 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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 Fall '11
 Wu
 Math, Linear Algebra, Determinant, Matrices, Invertible matrix, Row

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