深圳大学管理学院：《运筹学》课程教学资源（PPT课件讲稿）规划论（线性规划）运筹学3类 线性系统的解决 Solving Linear Systems

Chapter 2 Solving Linear Systems Matrix Definitions Matrix--- Rectangular array/ block of numbers 0 20 01 15 500 0 The size/order/dimension of a matrix (The numbers of RowS) by(x)(the numbers of COLUMNS)

Chapter 2 Solving Linear Systems • Matrix Definitions – Matrix--- Rectangular array/ block of numbers. – The size/order/dimension of a matrix: • (The numbers of ROWS) by(x) (the numbers of COLUMNS)                     − − − 0 500 0 1 1 0 0 1 15 1 0 20

ELEMENTS: individual numbers of matrix aj--an element of Row i and COLumn j SQURE matrix The numbers of rows= the numbers of COLUMNs DENTITY matrix: symbol TRANSPOSED matrix: Rows and columns of a matrix are switched A=25 456 36

– ELEMENTS: individual numbers of matrix – aij --- an element of ROW i and COLUMN j – SQURE matrix • The numbers of ROWS = the numbers of COLUMNS – IDENTITY matrix: symbol---I – TRANSPOSED matrix: Rows and columns of a matrix are switched –           =         = 6 5 4 3 2 1 4 5 6 1 2 3 t A A

Matrix Operations Addition Two same size matrices can be added ·C=A+B=B+A 101112 A=456 B=131415 789 61718 1+102+113+12(111315 C=4+135+146+15|=171921 7+168+179+18(232527

• Matrix Operations – Addition • Two same size matrices can be added. • C=A+B=B+A           =           + + + + + + + + + =           =           = 27 21 15 25 19 13 23 17 11 9 18 6 15 3 12 8 17 5 14 2 11 7 16 4 13 1 10 18 15 12 17 14 11 16 13 10 9 6 3 8 5 2 7 4 1 C A B

Multiplication Multiplication of a matrix by a scalar A=kA Example Mult plication of 2 Matrices Two Matrix can be multiplied if and only if--- The NUMBER OF COLUMNS OF THE FIRST MATRIX- The NUMBER OF ROWS OF THE SECOND MATRIX The Size of the resultant matrix the number of rows of the First matrix by the NUMBER OF COLUMNS OF THE SECOND MATRIX

– Multiplication • Multiplication of a Matrix by a Scalar – A=kA – Example • Multiplication of 2 Matrices – Two Matrix can be multiplied if and only if--- The NUMBER OF COLUMNS OF THE FIRST MATRIX = The NUMBER OF ROWS OF THE SECOND MATRIX – The Size of the resultant matrix --- the NUMBER OF ROWS OF THE FIRST MATRIX by the NUMBER OF COLUMNS OF THE SECOND MATRIX

E xample First Matrix Second Matrix Multipication Size Possible? A B AB (a)(2x2) (2x2) YES XL (b)(3x3) (3x2) YES (3x2) (c)(3x3) (2x3) NO (d)(5x5) YES (5X1)

• Example First Matrix Second Matrix Multipication Size Possible? A B AB (a）（2x2) (2x2) YES (2x2) (b）（3x3) (3x2) YES (3x2) (c）（3x3) (2x3) NO (d）（5x5) (5x1) YES (5x1)

· Notice that AB exists and so does ba with ba being(2x2) AB exists, BA does not exist as a (3x2)cannot be m ultiplied into a(3x3) ab does not exist, It's possible that ba exists How to calculate the elements of c=AB Exampl np e 123 A=456 B=11 789 68 C=AB= 67 266

• Notice that: – AB exists and so does BA with BA being (2x2) – AB exists, BA does not exist as a (3x2) cannot be multiplied into a (3x3) – AB does not exist, It’s possible that BA exists • How to calculate the elements of C=AB – Example           = =           =           = 266 67 68 12 11 10 9 6 3 8 5 2 7 4 1 C AB A B

56 B 34 78 1922 C= AB 4350 2334 C= BA 3146 AB≠B4

AB BA C BA C AB A B          = =         = =         =         = 31 46 23 34 43 50 19 22 7 8 5 6 3 4 1 2

A---mxn matrix identity matrix DIA=A DAI=A

– A---mxn matrix I=identity matrix »I A = A »A I = A

Matrix Inversion Only square matrices have the inverse but not all square matrices have inverses Scalar number. The inverse of matrix A is denoted by a-I The size of a-I is the same as a and AA=I=A-A Any matrix times its own inverse is just the appropriately sized identity matrix

– Matrix Inversion • Only Square matrices have the inverse but not all square matrices have inverses. • Scalar number: • • The inverse of matrix A is denoted by A-1 • The size of A-1 is the same as Aand • A A-1 = I = A-1 A • Any Matrix times its own inverse is just the appropriately sized identity matrix a a aa a a 1 1 1 1 1 = = = − − −

Matrix equality Two matrices are said to be equal if They are same size Corresponding elements in the two matrices are the same

– Matrix Equality • Two matrices are said to be equal if – They are same size – Corresponding elements in the two matrices are the same