Module 1: Matrix

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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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Nội dung Text: Module 1: Matrix

Natural Science Department Company LOGO 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 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 A3 5 = 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
Natural Science Department Company LOGO a11 = 300 300 420 360 450 600 A3 5 = 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 Company LOGO 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 � a11 a12 a13 ... a1j ... a1n � � a 21 a 22 a 23 ... a 2j ... a 2n � � � �... ... ... ... ... ... ... � A m×n =� �= � a � �a i1 a i2 a i3 ... a ij ... a in � � � ij 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
Natural Science Department Company LOGO Example: 1 �� 6 A = [ 100] B= �� C = [ 0 −3 100] �� 7 �� 0 �� 1 � 2 3 4� � 2 3 4 5� −5 4 −9 2 0 � � D= � � E=� � � 3 4 5 6� �4 3 7 8 2 � � � 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 would be written as A1 n = [ a11 a12 a13 ... a1n ] � aij � . or � � 1 n - Column-matrix: A matrix with only 1 column. A general column matrix would be written as �a11 � �a � Am 1 = � 21 � � aij � . �... � or � � m 1 � � �am1 � Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department Company LOGO A = [ 0] 1 �� B = �� 5 �� 1 �� 1 �� 6 �� C = �� 2 �� D= �� 7 �� 3 �� 0 �� Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department Company LOGO 2 4� � A = [ 100] B=� � 5 6� � 1 � 2 3 4� 0 0 2� � � � 2 3 4 5� C=� 1 � 2 3 � � D= � � 3 4 5 6� � 4 1 2� � � � � 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 general square matrix of order n would be written as �a11 a12 a13 ... a1n � �a a a 23 ... a 2n � � 21 22 � An×n =�a 31 a 32 a 33 ... a 3n � � a � . � � or �� ij n×n �... ... ... ... ... � � �a n1 a n2 a n3 ... a nn � � a11 ,a 22 ,a 33 ,...,a ii ,...,a nn: main diagonal of A. Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department Company LOGO 2 4� � A = [ 100] B=� � 5 6� � 1 � 2 3 4� 0 −2 3� � � � 2 3 4 5� C=� 1 � 2 9 � � D= � � 3 4 5 6� � 4 8 6� � � � � 4 � 5 6 7� Chapter 1: Matrix, Determinant, System of linear equations Module 1: Matrix
Natural Science Department Company LOGO I1 = [ 1] 1 0� � I2 = � � 0 1� � 1 � 0 0 0� 1 0 0� � � � � 0 1 0 0� I3 = � 0 1 0� I4 = � �;... � 0 0 1 0� � 0 0 1� � � � � 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 general matrix unit of order n would be written as 1 � 0 ... 0� �0 1 ... 0� � In = � � ... ... ... ...� � � �0 0 ... 1� 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 zero matrix. 0 0� � 0 0 0� � A = [ 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
Natural Science Department Company LOGO 3. Operations on matrices - Two matrices are defined to be equal if they have the same size and the corresponding entries are equal. � a �� ij � = � b �ij � � � 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
Natural Science Department Company LOGO 3. Operations on matrices - Transposition: T Let A is any mxn matrix, the transpose of A, denoted by 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