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Example 1:
The following rectangular array describes the profit (milions dollar)
of 3 branches in 5 years:
2008
2009
2010
2011
2012
I
300
420
360
450
600
II
310
250
300
210
340
III
600
630
670
610
700
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Duy Tân University
Natural Science Department
Module 1:
MATRIX
Lecturer: Thân Thị Quỳnh Dao
Company LOGO
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Company LOGO
1. Definition
- A matrix is a rectangular array of numbers. The numbers in
the array are called the entries in the matrix.
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Company LOGO
300 420 360 450 600
310 250 300 210 340
600 630 670 610 700
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Company LOGO
300 420 360 450 600
310 250 300 210 340
A = 3 5A (cid:0) =
600 630 670 610 700
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Company LOGO
1. Definition
- A matrix is a rectangular array of numbers. The numbers in
the array are called the entries in the matrix.
- We use the capital letters to denote matrices such as A, B, C ...
- The size of matrix is described in terms of the number of rows
and columns it contains.
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
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a =
300
11
300 420 360 450 600
3 5A (cid:0) =
310 250 300 210 340
a =
210
24
600 630 670 610 700
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Company LOGO
1. Definition
- Let m,n are positive integers. A general mxn matrix is a
a
a
a
a
...
1j
1n
13
12
11
a
a
a
a
...
...
...
2j ...
22 ...
23 ...
2n ...
...
A
m×n
a� �= � � ij m×n
...
...
a
a
a
a
in ...
ij ...
... ... a
... ... a
i3 ... a
i2 ... a
mn
m1
m3
m2
mj
� � � � � � � � � �
a � � a � 21 � ... = � a � i1 � ... � a � � ija : the entry occurs in row i and column j.
rectangular array of number with m rows and n columns as ...
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Company LOGO
Example:
[
]
[
]
C =
0
3 100
A =
B
100
1 �� �� 6 ��= �� 7 �� 0 ��
-
=
=
E
D
9 2 0 8 2 7
5 4 � � 4 3 �
� � �
1 2 3 4 � � 2 3 4 5 � � 3 4 5 6 � 4 5 6 7 �
� � � � � �
- -
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Company LOGO
[
]
A =
3 5
[
]
B =
7 9 2 4
[
]
C =
2 5 7 8 2 3 0
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Company LOGO
2. Some special matrices
- Row-matrix: A matrix with only 1 row. A general row matrix
]
(cid:0) =
...
.
n
a 11
a 12
A 1
a 13
a n 1
� �� � a ij
n
1
would be written as [ or (cid:0)
- Column-matrix: A matrix with only 1 column. A general
.
column matrix would be written as
A m
1
� �� � a ij m
1
a � � 11 � � a � �= 21 � � ... � � a � � m 1
or (cid:0) (cid:0)
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Company LOGO
[
A =
]0
B
1 ��= �� 5 ��
C
D
1 �� ��= 2 �� ���� 3
1 �� �� 6 ��= �� 7 �� 0 ��
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Company LOGO
=
B
[
]
A =
100
2 4 � � � � 5 6 � �
=
=
D
C
0 0 2 � � 1 2 3 � � 4 1 2 �
� � � � �
1 2 3 4 � � 2 3 4 5 � � 3 4 5 6 � 4 5 6 7 �
� � � � � �
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Company LOGO
2. Some special matrices
- Square matrix of order n: A matrix with n rows, n columns.
a
a
...
a
12
13
1n
a
a
... a
22
23
2n
=
A
a
a
... a
n×n
.
A general square matrix of order n would be written as
ija� �� �
n×n
32 ...
33 ...
3n ...
...
a
a
... a
n2
n3
nn
a � 11 � a � 21 � a 31 � ... � � a � n1
� � � � � � � �
or
a ,a ,a ,...,a ,...,a : 11 nn
22
33
ii
main diagonal of A.
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Company LOGO
=
B
[
]
A =
100
2 4 � � � � 5 6 � �
=
=
D
C
2 3 9 2 6 8
-� 0 � 1 � � 4 �
� � � � �
1 2 3 4 � � 2 3 4 5 � � 3 4 5 6 � 4 5 6 7 �
� � � � � �
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Company LOGO
[
=
I =
] 1
1
I
2
1 0 � � � � 0 1 � �
=
=
I
3
I
;...
4
1 0 0 � � 0 1 0 � � 0 0 1 �
� � � � �
1 0 0 0 � � 0 1 0 0 � � 0 0 1 0 � 0 0 0 1 �
� � � � � �
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Company LOGO
2. Some special matrices
- Matrix unit of order n: A square matrix of order n whose all
entris on the main diagonal are 1 and the others are 0. A
0 ... 0
1 ... 0
=
I
n
... ... ... 0 ... 1
1 � � 0 � � ... � 0 �
� � � � � �
general matrix unit of order n would be written as
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Company LOGO
2. Some special matrices
- Zero matrix: a matrix, all of whose entries are zero, is called
0 0 0
=
=
zero matrix.
[
A =
]0
B
C
0 0 0
0 0 � � � ; � � � 0 0 � � �
� � �
0 0 0 0 0
=
D
0 0 0 0 0 ;...
0 0 0 0 0
0 0 0 � � 0 0 0 ; � � 0 0 0 �
� � � � = E � � � � � �
� � � � �
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Company LOGO
3. Operations on matrices
=
�
" = i
n
= m j ,
1,
1,
;
a ij
b ij
b ij
- Two matrices are defined to be equal if they have
m n
(cid:0) (cid:0) the same size and the corresponding entries are equal. � � � �= a � � � � ij m n
=
=
=
A
B
C
1 0 3 � � 2 4 1 �
� ; � �
1 0 3 � � x 2 1 �
� ; � �
1 0 � � � � 2 4 � �
Example: Find x such that A = B, B = C?
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Company LOGO
3. Operations on matrices
TA
- Transposition:
Let A is any mxn matrix, the transpose of A, denoted by
is defined to be the nxm matrix that results from interchanging
the rows and the columns of A.
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Company LOGO
3. Operations on matrices
=
- Addition and subtraction:
b ij
(cid:0) (cid:0)
� � � � � a b a � � � � � ij ij ij
m n
m n
� � m n
(cid:0) (cid:0) (cid:0)
=
=
=
A
B
C
1 0 3 � � 2 4 1 �
� ; � �
3 4 5 � � -� 1 0 2
� ; � �
1 0 � � � � 2 4 � �
Example: Find (if any): A + B, A – B, B + C?
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Company LOGO
3. Operations on matrices
c a ij
ca ij
- Scalar multiples: let c is real number
= � � � � � � � � m n
m n
(cid:0) (cid:0)
=
A
1 0 3 � � 2 4 1 �
� � �
Example: Find 3A?
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department Natural Science Department
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3. Operations on matrices
=
A
3 1 ;
Example: Find: 2A + 3B – I3 , with:
2 0 2 0
0 0 0 2 1 4 3 0 1
1 � � 2 � � 1 �
� � � � = B � � � � � �
� � � � �
- -
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department Natural Science Department
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3. Operations on matrices
n
=
=
- Multiplying matrices:
b
ij
ij
a b ik
kj
� � � �(cid:0) a � � � � n×p
m×n
k = 1
� c � ij �
� � � m×p
(cid:0)
=
A
1 0 3 � � 2 4 1 �
1
1 �� � �� = B 2 ; � �� � ����
Example: Find AB?
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department Natural Science Department
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