Solving Linear Systems - Addition

Chapter 2 Solving Linear SystemsMatrixDefinitions- Matrix--- Rectangular array/ block of numbers(10(0-20)01-1550000-11- The size/order/dimension of a matrix·(Thenumbers ofROWS)by(x)(thenumbers ofCOLUMNS)

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- aij -- an element of ROW i and COLUMN j-SOUREmatrix·ThenumbersofROWS=thenumbersofCOLUMNS- IDENTITY matrix: symbol---I- TRANSPOSED matrix: Rows and columns ofamatrix are switched42315At=12A=54636

− 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(123(10 11 12)B=614 154513A=917188116(11 )15(1+102+113+12）1321C=|4+13 5+14 6+15|=17 19(23 25 277+16 8+17 9+18）

• 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·Multiplication of2Matrices-Two Matrixcanbemultipliedif andonlyif---TheNUMBER OFCOLUMNSOFTHEFIRSTMATRIX=TheNUMBEROFROWSOFTHESECONDMATRIX-The Size ofthe resultantmatrix ---theNUMBEROFROWSOFTHEFIRSTMATRIXbytheNUMBEROFCOLUMNSOFTHESECONDMATRIX

− 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

·ExampleSizeFirst Matrix Second MatrixMultipicationPossible?ABAB(a)(2x2)YES(2x2)(2x2)(b)(3x3)(3x2)YES(3x2)(c)(3x3)NO(2x3)(d)(5x5)YES(5xl)(5xl)

• 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)