Chapter 2 Solving Linear Systems Matrix Definitions Matrix

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)

– 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 –

• Matrix Operations – Addition • Two same size matrices can be added. • C=A+B=B+A

– Multiplication • Multiplication of a Matrix by a Scalar – A=k. A – 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）（2 x 2) (b）（3 x 3) (c）（3 x 3) (d）（5 x 5) B (2 x 2) (3 x 2) (2 x 3) (5 x 1) Multipication Possible? AB YES NO YES Size (2 x 2) (3 x 2) (5 x 1)

• Notice that: – AB exists and so does BA with BA being (2 x 2) – AB exists, BA does not exist as a (3 x 2) cannot be multiplied into a (3 x 3) – AB does not exist, It’s possible that BA exists • How to calculate the elements of C=AB – Example

– 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: • • • 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

– 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 – 20 q = 0 0 TR + 1 TC – 25 q = 500 1 TR – 1 TC + 0 q = 0 – Let

– Ax=b – Example A-1 Ax= A-1 b I x= A-1 b

– 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-1 b Use the matrix equality rule to find the elements of x Give the business interpretation of x