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)