1. Tree Diagram 2. Medical Example 3. Quality Control Example 1.
Example 1:
description
Transcript of Example 1:
Company
LOGO
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
Company
LOGO
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
Company
LOGO
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.
Company
LOGO
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
LOGO
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
Company
LOGO
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.
Company
LOGO
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
Company
LOGO
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
Company
LOGO
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
Company
LOGO
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
Company
LOGO
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
Company
LOGO
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
Company
LOGO
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
Company
LOGO
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 :
Company
LOGO
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
Company
LOGO
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
Company
LOGO
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
Company
LOGO
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
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
Company
LOGO
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
Company
LOGO
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
Company
LOGO
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
Company
LOGO
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
Company
LOGO
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
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
Company
LOGO
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
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
Company
LOGO
Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department
;
Natural Science Department