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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
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
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Module 1:
MATRIX
Duy Tân University
Lecturer: Thân Thị Quỳnh Dao
Natural Science Department
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
Natural Science Department
1. Definition
- A matrix is a rectangular array of numbers. The numbers in
the array are called the entries in the matrix.
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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
300 420 360 450 600
310 250 300 210 340
600 630 670 610 700
Company
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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
300 420 360 450 600
310 250 300 210 340
600 630 670 610 700
A 3 5A
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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
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.
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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
3 5A
11 300a
24 210a
300 420 360 450 600
310 250 300 210 340
600 630 670 610 700
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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
11 12 13 1j 1n
21 22 23 2j 2n
m×ni1 i2 i3 ij in
m1 m2 m3 mj mn
a a a ... a ... a
a a a ... a ... a
... ... ... ... ... ... ...A
a a a ... a ... a
... ... ... ... ... ... ...
a a a ... a ... a
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
the entry occurs in row i and column j.ija :
ij m×na
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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
Example:
100A 0 3 100C
1
6
7
0
B
1 2 3 4
2 3 4 5
3 4 5 6
4 5 6 7
D
5 4 9 2 0
4 3 7 8 2E
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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
7 9 2 4B
2 5 7 8 2 3 0C
3 5A
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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
2. Some special matrices
- Row-matrix: A matrix with only 1 row. A general row matrix
would be written as
1 11 12 13 1...n nA a a a a
- Column-matrix: A matrix with only 1 column. A general
column matrix would be written as
11
211
1
...m
m
a
aA
a
or 1.ij n
a
or 1.ij m
a
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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
1
2
3
C
1
6
7
0
D
1
5B
0A
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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
100A
0 0 2
1 2 3
4 1 2
C
2 4
5 6B
1 2 3 4
2 3 4 5
3 4 5 6
4 5 6 7
D
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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
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
11 12 13 1n
21 22 23 2n
n×n 31 32 33 3n
n1 n2 n3 nn
a a a ... a
a a a ... a
a a a ... a
... ... ... ... ...
a a a ... a
A
or n×n.ija
main diagonal of A.11 22 33 ii nna ,a ,a ,...,a ,...,a :
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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
100A
0 2 3
1 2 9
4 8 6
C
2 4
5 6B
1 2 3 4
2 3 4 5
3 4 5 6
4 5 6 7
D
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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
1 1I 2
1 0
0 1I
3
1 0 0
0 1 0
0 0 1
I
4
1 0 0 0
0 1 0 0;...
0 0 1 0
0 0 0 1
I
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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
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
n
1 0 ... 0
0 1 ... 0I
... ... ... ...
0 0 ... 1
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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.
0A0 0 0 0 0
;0 0 0 0 0
B C
0 0 0 0 0 0 0 0
0 0 0 ; 0 0 0 0 0 ;...
0 0 0 0 0 0 0 0
D E
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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
3. Operations on matrices
- Two matrices are defined to be equal if they have
the same size and the corresponding entries are equal.
; 1, , 1,ij ij ij ijm n m na b a b i m j n
Example: Find x such that A = B, B = C?
1 0 3;
2 4 1A
1 0 3;
2 1B
x
1 0
2 4C
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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
3. Operations on matrices
- 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.
TA
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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
3. Operations on matrices
- Addition and subtraction:
ij ij ij ijm n m n m na b a b
Example: Find (if any): A + B, A – B, B + C?
1 0 3;
2 4 1A
3 4 5;
1 0 2B
1 0
2 4C
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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
3. Operations on matrices
- Scalar multiples: let c is real number
ij ijm n m nc a ca
Example: Find 3A?
1 0 3
2 4 1A
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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
Natural Science Department
3. Operations on matrices
Example: Find: 2A + 3B – I3 , with:
1 2 3 0 0 0
2 0 1 ; 2 1 4
1 2 0 3 0 1
A B
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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
Natural Science Department
3. Operations on matrices
- Multiplying matrices:
ij ij ij ik kjm×n n×p1 m×p
a b c a bn
k
Example: Find AB?
11 0 3
; 22 4 1
1
A B
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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
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Natural Science Department