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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix

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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

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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

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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.

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Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix

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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

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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

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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.

- 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

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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

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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

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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

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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

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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

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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

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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

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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

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

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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

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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

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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

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

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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.

; 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

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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

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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

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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

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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

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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

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