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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
CompanyThị Quỳnh Dao
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Chapter 1: Matrix, Determinant, System of linear equations
Module 1: Matrix
Natural Science Department
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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
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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
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A35 
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
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
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A35 
a11  300
300
420
360
450
600
310
250
300
210
340
600
630
670
610
700
a24  210
Chapter 1: Matrix, Determinant, System of linear equations
Module 1: Matrix
Natural Science Department
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1. Definition
- Let m,n are positive integers. A general mxn matrix is a
rectangular array of number with m rows and n columns as
A m×n
 a11 a12 a13 ... a1j ... a1n 
a

a
a
...
a
...
a
21
22
23
2j
2n 

 ...
...
... ... ... ... ... 

  a ij 
 a i1 a i2 a i3 ... a ij ... a in    m×n
 ...
...
... ... ... ... ... 


a m1 a m2 a m3 ... a mj ... a mn 
a ij : the entry occurs in row i and column j.
Chapter 1: Matrix, Determinant, System of linear equations
Module 1: Matrix
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Example:
1 
6
B 
7 
 
0
A  100
1
2
D
3

4
2
3
4
5
3
4
5
6
4
5 
6

7
C  0 3 100
 5 4 9 2 0 
E

4
3
7
8
2


Chapter 1: Matrix, Determinant, System of linear equations
Module 1: Matrix
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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
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2. Some special matrices
- Row-matrix: A matrix with only 1 row. A general row matrix
would be written as
A1n   a11 a12
a13 ... a1n 
or
aij  .
1n
- Column-matrix: A matrix with only 1 column. A general
column matrix would be written as
 a11 
a 
21 

Am1 
 ... 
 
 am1 
or
aij  .
m1
Chapter 1: Matrix, Determinant, System of linear equations
Module 1: Matrix
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A  0
1
B 
5 
1 


C  2
 3 
1 
6
D 
7 
 
0
Chapter 1: Matrix, Determinant, System of linear equations
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A  100
0 0 2
C  1 2 3 
 4 1 2
 2 4
B

5 6
1
2
D
3

4
2
3
4
5
Chapter 1: Matrix, Determinant, System of linear equations
3
4
5
6
4

5
6

7
Module 1: Matrix
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2. Some special matrices
- Square matrix of order n: A matrix with n rows, n columns.
A general square matrix of order n would be written as
An×n
 a11 a12
a
a
21
22

  a 31 a 32

 ... ...
a n1 a n2
a13 ... a1n 
a 23 ... a 2n 
a 33 ... a 3n 

... ... ... 
a n3 ... a nn 
or
aij  .
n×n
a11 ,a 22 ,a 33 ,...,a ii ,...,a nn: main diagonal of A.
Chapter 1: Matrix, Determinant, System of linear equations
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A  100
0 2 3
C  1 2 9 
 4 8 6
 2 4
B

5 6
1
2
D
3

4
2
3
4
5
Chapter 1: Matrix, Determinant, System of linear equations
3
4
5
6
4

5
6

7
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I1  1
1 0 0 


I 3  0 1 0 
0 0 1 
1 0 
I2  

0 1 
1
0
I4  
0

0
0
1
0
0
Chapter 1: Matrix, Determinant, System of linear equations
0
0
1
0
0
0 
;...
0

1
Module 1: Matrix
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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
general matrix unit of order n would be written as
1
0
In  
...

0
0
1
...
0
...
...
...
...
0

0
...

1
Chapter 1: Matrix, Determinant, System of linear equations
Module 1: Matrix
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2. Some special matrices
- Zero matrix: a matrix, all of whose entries are zero, is called
zero matrix.
A  0
0 0 
0 0 0 
B
;C


0 0 
0 0 0 
0 0 0
0 0 0 0 0




D  0 0 0 ; E  0 0 0 0 0 ;...
0 0 0
0 0 0 0 0
Chapter 1: Matrix, Determinant, System of linear equations
Module 1: Matrix
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3. Operations on matrices
- Two matrices are defined to be equal if they have
the same size and the corresponding entries are equal.
aij   bij   aij  bij ; i  1, m, j  1, n
mn
mn
Example: Find x such that A = B, B = C?
 1 0 3
1 0 3
1 0 
A
; B
; C



2
4
1
2
x
1




 2 4
Chapter 1: Matrix, Determinant, System of linear equations
Module 1: Matrix
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3. Operations on matrices
- Transposition:
Let A is any mxn matrix, the transpose of A, denoted by
T
A
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
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3. Operations on matrices
- Addition and subtraction:
aij   bij   aij  bij 
mn
mn
mn
Example: Find (if any): A + B, A – B, B + C?
 1 0 3
 3 4 5
1 0 
A
; B
; C



 2 4 1
 1 0 2 
 2 4
Chapter 1: Matrix, Determinant, System of linear equations
Module 1: Matrix
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3. Operations on matrices
- Scalar multiples: let c is real number
c aij 
mn
 caij 
mn
Example: Find 3A?
 1 0 3
A

2
4
1


Chapter 1: Matrix, Determinant, System of linear equations
Module 1: Matrix
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3. Operations on matrices
Example: Find: 2A + 3B – I3 , with:
 1 2 3
0 0 0 
A   2 0 1  ; B   2 1 4 
 1 2 0
 3 0 1 
Chapter 1: Matrix, Determinant, System of linear equations
Module 1: Matrix
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3. Operations on matrices
- Multiplying matrices:
a ij    bij 
m×n
n×p
n


 cij   a ik bkj 
k 1

 m×p
Example: Find AB?
1 0 3
A
;

 2 4 1
1 


B  2
1 
Chapter 1: Matrix, Determinant, System of linear equations
Module 1: Matrix
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;
Chapter 1: Matrix, Determinant, System of linear equations
Module 1: Matrix