2INTRODUCTORY MATHEMATICAL ANALYSIS 0. Review of AlgebraApplications and More AlgebraFunctions and GraphsLines, Parabolas, and SystemsExponential and Logarithmic FunctionsMathematics of FinanceMatrix AlgebraLinear ProgrammingIntroduction to Probability and Statistics
3INTRODUCTORY MATHEMATICAL ANALYSIS Additional Topics in ProbabilityLimits and ContinuityDifferentiationAdditional Differentiation TopicsCurve SketchingIntegrationMethods and Applications of IntegrationContinuous Random VariablesMultivariable Calculus
4Chapter Objectives Concept of a matrix. Special types of matrices. Chapter 6: Matrix AlgebraChapter ObjectivesConcept of a matrix.Special types of matrices.Matrix addition and scalar multiplication operations.Express a system as a single matrix equation using matrix multiplication.Matrix reduction to solve a linear system.Theory of homogeneous systems.Inverse matrix.Use a matrix to analyze the production of sectors of an economy.
5Chapter Outline 6.1) Matrices Chapter 6: Matrix AlgebraChapter OutlineMatricesMatrix Addition and Scalar MultiplicationMatrix MultiplicationSolving Systems by Reducing MatricesSolving Systems by Reducing Matrices (continued)InversesLeontief’s Input—Output Analysis6.1)6.2)6.3)6.4)6.5)6.6)6.7)
6Chapter 6: Matrix Algebra 6.1 MatricesA matrix consisting of m horizontal rows and n vertical columns is called an m×n matrix or a matrix of size m×n.For the entry aij, we call i the row subscript and j the column subscript.
7a. The matrix has size . b. The matrix has size . Chapter 6: Matrix Algebra6.1 MatricesExample 1 – Size of a Matrixa. The matrix has size .b. The matrix has size .c. The matrix has size .d. The matrix has size
8Example 3 – Constructing Matrices Equality of Matrices Chapter 6: Matrix Algebra6.1 MatricesExample 3 – Constructing MatricesEquality of MatricesMatrices A = [aij ] and B = [bij] are equal if they have the same size and aij = bij for each i and j.Transpose of a MatrixA transpose matrix is denoted by AT.If , find .Solution:Observe that
96.2 Matrix Addition and Scalar Multiplication Chapter 6: Matrix Algebra6.2 Matrix Addition and Scalar MultiplicationExample 1 – Matrix AdditionMatrix AdditionSum A + B is the m × n matrix obtained by adding corresponding entries of A and B.a.b is impossible as matrices are not of the samesize.
10Demand for the consumers is For the industries is Chapter 6: Matrix Algebra6.2 Matrix Addition and Scalar MultiplicationExample 3 – Demand Vectors for an EconomyDemand for the consumers isFor the industries isWhat is the total demand for consumers and the industries?Solution:Total:
11Scalar Multiplication Properties of Scalar Multiplication: Chapter 6: Matrix Algebra6.2 Matrix Addition and Scalar MultiplicationScalar MultiplicationProperties of Scalar Multiplication:Subtraction of MatricesProperty of subtraction is
12a. b. Example 5 – Matrix Subtraction Chapter 6: Matrix Algebra 6.2 Matrix Addition and Scalar MultiplicationExample 5 – Matrix Subtractiona.b.
136.3 Matrix Multiplication Chapter 6: Matrix Algebra6.3 Matrix MultiplicationExample 1 – Sizes of Matrices and Their ProductAB is the m× p matrix C whose entry cij is given byA = 3 × 5 matrixB = 5 × 3 matrixAB = 3 × 3 matrix but BA = 5 × 5 matrix.C = 3 × 5 matrixD = 7 × 3 matrixCD = undefined but DC = 7 × 5 matrix.
15Given the price and the quantities, calculate the total cost. Chapter 6: Matrix Algebra6.3 Matrix MultiplicationExample 5 – Cost VectorGiven the price and the quantities, calculate the total cost.Solution:The cost vector is
16If compute ABC in two ways. Solution 1: Solution 2: Chapter 6: Matrix Algebra6.3 Matrix MultiplicationExample 7 – Associative PropertyIfcompute ABC in two ways.Solution 1: Solution 2:Note that A(BC) = (AB)C.
17Find QRC when Solution: Example 9 – Raw Materials and Cost Chapter 6: Matrix Algebra6.3 Matrix MultiplicationExample 9 – Raw Materials and CostFind QRC whenSolution:
18Solution: Example 11 – Matrix Operations Involving I and O If Chapter 6: Matrix Algebra6.3 Matrix MultiplicationExample 11 – Matrix Operations Involving I and OIfcompute each of the following.Solution:
19in matrix form by using matrix multiplication. Chapter 6: Matrix Algebra6.3 Matrix MultiplicationExample 13 – Matrix Form of a System Using Matrix MultiplicationWrite the systemin matrix form by using matrix multiplication.Solution:Ifthen the single matrix equation is
206.4 Solving Systems by Reducing Matrices Chapter 6: Matrix Algebra6.4 Solving Systems by Reducing MatricesElementary Row OperationsInterchanging two rows of a matrixMultiplying a row of a matrix by a nonzero numberAdding a multiple of one row of a matrix to a different row of that matrix
21Properties of a Reduced Matrix All zero-rows at the bottom. Chapter 6: Matrix Algebra6.4 Solving Systems by Reducing MatricesProperties of a Reduced MatrixAll zero-rows at the bottom.For each nonzero-row, leading entry is 1 and the rest zeros.Leading entry in each row is to the right of the leading entry in any row above it.
22Chapter 6: Matrix Algebra 6.4 Solving Systems by Reducing MatricesExample 1 – Reduced MatricesFor each of the following matrices, determine whether it is reduced or not reduced.Solution:a. Not reduced b. Reducedc. Not reduced d. Reducede. Not reduced f. Reduced
23By using matrix reduction, solve the system Chapter 6: Matrix Algebra6.4 Solving Systems by Reducing MatricesExample 3 – Solving a System by ReductionBy using matrix reduction, solve the systemSolution:Reducing the augmented coefficient matrix of the system,We have
24Using matrix reduction, solve Chapter 6: Matrix Algebra6.4 Solving Systems by Reducing MatricesExample 5 – Parametric Form of a SolutionUsing matrix reduction, solveSolution:Reducing the matrix of the system,We have and x4 takes on any real value.
256.5 Solving Systems by Reducing Matrices (continued) Chapter 6: Matrix Algebra6.5 Solving Systems by Reducing Matrices (continued)Example 1 – Two-Parameter Family of SolutionsUsing matrix reduction, solveSolution:The matrix is reduced toThe solution is
26is called a homogeneous system if c1 = c2 = … = cm = 0. Chapter 6: Matrix Algebra6.5 Solving Systems by Reducing Matrices (Continue)The systemis called a homogeneous system if c1 = c2 = … = cm = 0.The system is non-homogeneous if at least one of the c’s is not equal to 0.Concept for number of solutions:k < n infinite solutionsk = n unique solution
272 equations (k), homogeneous system, 3 unknowns (n). Chapter 6: Matrix Algebra6.5 Solving Systems by Reducing Matrices (Continue)Example 3 – Number of Solutions of a Homogeneous SystemDetermine whether the system has a unique solution or infinitely many solutions.Solution:2 equations (k), homogeneous system, 3 unknowns (n).The system has infinitely many solutions.
28Chapter 6: Matrix Algebra 6.6 InversesExample 1 – Inverse of a MatrixWhen matrix CA = I, C is an inverse of A and A is invertible.Let and Determine whether C isan inverse of A.Solution:Thus, matrix C is an inverse of A.
29Method to Find the Inverse of a Matrix When matrix is reduced, , Chapter 6: Matrix Algebra6.6 InversesExample 3 – Determining the Invertibility of a MatrixMethod to Find the Inverse of a MatrixWhen matrix is reduced, ,If R = I, A is invertible and A−1 = B.If R I, A is not invertible.Determine if is invertible.Solution: We haveMatrix A is invertible where
30Solve the system by finding the inverse of the coefficient matrix. Chapter 6: Matrix Algebra6.6 InversesExample 5 – Using the Inverse to Solve a SystemSolve the system by finding the inverse of the coefficient matrix.Solution:We haveFor inverse,The solution is given by X = A−1B:
316.7 Leontief’s Input-Output Analysis Chapter 6: Matrix Algebra6.7 Leontief’s Input-Output AnalysisExample 1 – Input-Output AnalysisEntries are called input–output coefficients.Use matrices to show inputs and outputs.Given the input–output matrix,suppose final demand changes to be 77 for A, 154 for B, and 231 for C. Find the output matrix for the economy. (The entries are in millions of dollars.)
32Divide entries by the total value of output to get A: Chapter 6: Matrix Algebra6.7 Leontief’s Input-Output AnalysisExample 1 – Input-Output AnalysisSolution:Divide entries by the total value of output to get A:Final-demand matrix:Output matrix is