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Chapter 2 Solving Linear Systems • Matrix Definitions – Matrix--- Rectangular array/ block of numbers. 1 0 20 0 1 15 1 1 0 0 500 0 – The size/order/dimension of a matrix: • (The numbers of ROWS) by(x) (the numbers of COLUMNS) – 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 – 1 4 1 2 3 A 4 5 6 A 2 5 3 6 t • Matrix Operations – Addition • Two same size matrices can be added. • C=A+B=B+A 1 2 3 A 4 5 6 7 8 9 10 11 12 B 13 14 15 16 17 18 1 10 2 11 3 12 11 13 15 C 4 13 5 14 6 15 17 19 21 7 16 8 17 9 18 23 25 27 – 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 • Example First Matrix Second Matrix A (a)(2x2) (b)(3x3) (c)(3x3) (d)(5x5) B (2x2) (3x2) (2x3) (5x1) Multipication Possible? AB YES YES NO YES Size (2x2) (3x2) (5x1) • 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 1 2 3 A 4 5 6 7 8 9 68 C AB 67 266 10 B 11 12 1 A 3 2 5 B 4 7 19 22 C AB 43 50 23 34 C BA 31 46 AB BA 6 8 – A---mxn matrix » IA=A » AI =A I=identity matrix – Matrix Inversion • Only Square matrices have the inverse but not all square matrices have inverses. • Scalar number: aa 1 1 a 1a • 1 a a The inverse of matrix A is denoted by A-1 The size of A-1 is the same as A and A A-1 = I = A-1 A Any Matrix times its own inverse is just the appropriately sized identity matrix 1 • • • • – Matrix Equality • Two matrices are said to be equal if – They are same size – Corresponding elements in the two matrices are the same • Break-Even Model in Matrix Algebra terms – Break-even model in linear equations 1 TR + 0 TC – 20q = 0 0 TR + 1 TC – 25q = 500 1 TR – 1 TC + 0q = 0 – Let 1 0 20 TR 0 A 0 1 15 x TC b 500 1 1 0 q 0 3x3 3 x1 3 x1 A3x3 x3 x1 b3 x1 – Example – Ax=b A-1 Ax= A-1 b I x= A-1 b x= A-1 b 4 4 0 2000 -3 1 x A b -3 4 3 500 2000 - 0.2 0.2 0.2 0 100 – Modelling Steps • Set up the system of linear equations • Decide upon an order in which to express the unknowns • The unknowns on the LHS of the equations • Identify the following 3 matrices – A: Square matrix of coefficients relating to the unknowns – x: the matrix of unknows – b: the matrix of RHS constants • • • • Find matrix inverse A-1 of A Perform the matrix multiplication A-1b Use the matrix equality rule to find the elements of x Give the business interpretation of x