Chapter 6 Matrix Algebra

INTRODUCTORY MATHEMATICAL ANALYSIS 0. Review of Algebra Applications and More Algebra Functions and Graphs Lines, Parabolas, and Systems Exponential and Logarithmic Functions Mathematics of Finance Matrix Algebra Linear Programming Introduction to Probability and Statistics

INTRODUCTORY MATHEMATICAL ANALYSIS Additional Topics in Probability Limits and Continuity Differentiation Additional Differentiation Topics Curve Sketching Integration Methods and Applications of Integration Continuous Random Variables Multivariable Calculus

Chapter Objectives Concept of a matrix. Special types of matrices. Chapter 6: Matrix Algebra Chapter Objectives Concept 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.

Chapter Outline 6.1) Matrices Chapter 6: Matrix Algebra Chapter Outline Matrices Matrix Addition and Scalar Multiplication Matrix Multiplication Solving Systems by Reducing Matrices Solving Systems by Reducing Matrices (continued) Inverses Leontief’s Input—Output Analysis 6.1) 6.2) 6.3) 6.4) 6.5) 6.6) 6.7)

Chapter 6: Matrix Algebra 6.1 Matrices A 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.

a. The matrix has size . b. The matrix has size . Chapter 6: Matrix Algebra 6.1 Matrices Example 1 – Size of a Matrix a. The matrix has size . b. The matrix has size . c. The matrix has size . d. The matrix has size .

Example 3 – Constructing Matrices Equality of Matrices Chapter 6: Matrix Algebra 6.1 Matrices Example 3 – Constructing Matrices Equality of Matrices Matrices A = [aij ] and B = [bij] are equal if they have the same size and aij = bij for each i and j. Transpose of a Matrix A transpose matrix is denoted by AT. If , find . Solution: Observe that .

6.2 Matrix Addition and Scalar Multiplication Chapter 6: Matrix Algebra 6.2 Matrix Addition and Scalar Multiplication Example 1 – Matrix Addition Matrix Addition Sum 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 same size.

Demand for the consumers is For the industries is Chapter 6: Matrix Algebra 6.2 Matrix Addition and Scalar Multiplication Example 3 – Demand Vectors for an Economy Demand for the consumers is For the industries is What is the total demand for consumers and the industries? Solution: Total:

Scalar Multiplication Properties of Scalar Multiplication: Chapter 6: Matrix Algebra 6.2 Matrix Addition and Scalar Multiplication Scalar Multiplication Properties of Scalar Multiplication: Subtraction of Matrices Property of subtraction is

a. b. Example 5 – Matrix Subtraction Chapter 6: Matrix Algebra 6.2 Matrix Addition and Scalar Multiplication Example 5 – Matrix Subtraction a. b.

6.3 Matrix Multiplication Chapter 6: Matrix Algebra 6.3 Matrix Multiplication Example 1 – Sizes of Matrices and Their Product AB is the m× p matrix C whose entry cij is given by A = 3 × 5 matrix B = 5 × 3 matrix AB = 3 × 3 matrix but BA = 5 × 5 matrix. C = 3 × 5 matrix D = 7 × 3 matrix CD = undefined but DC = 7 × 5 matrix.

Example 3 – Matrix Products Chapter 6: Matrix Algebra 6.3 Matrix Multiplication Example 3 – Matrix Products a. b. c. d.

Given the price and the quantities, calculate the total cost. Chapter 6: Matrix Algebra 6.3 Matrix Multiplication Example 5 – Cost Vector Given the price and the quantities, calculate the total cost. Solution: The cost vector is

If compute ABC in two ways. Solution 1: Solution 2: Chapter 6: Matrix Algebra 6.3 Matrix Multiplication Example 7 – Associative Property If compute ABC in two ways. Solution 1: Solution 2: Note that A(BC) = (AB)C.

Find QRC when Solution: Example 9 – Raw Materials and Cost Chapter 6: Matrix Algebra 6.3 Matrix Multiplication Example 9 – Raw Materials and Cost Find QRC when Solution:

Solution: Example 11 – Matrix Operations Involving I and O If Chapter 6: Matrix Algebra 6.3 Matrix Multiplication Example 11 – Matrix Operations Involving I and O If compute each of the following. Solution:

in matrix form by using matrix multiplication. Chapter 6: Matrix Algebra 6.3 Matrix Multiplication Example 13 – Matrix Form of a System Using Matrix Multiplication Write the system in matrix form by using matrix multiplication. Solution: If then the single matrix equation is

6.4 Solving Systems by Reducing Matrices Chapter 6: Matrix Algebra 6.4 Solving Systems by Reducing Matrices Elementary Row Operations Interchanging two rows of a matrix Multiplying a row of a matrix by a nonzero number Adding a multiple of one row of a matrix to a different row of that matrix

Properties of a Reduced Matrix All zero-rows at the bottom. Chapter 6: Matrix Algebra 6.4 Solving Systems by Reducing Matrices Properties of a Reduced Matrix All 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.

Chapter 6: Matrix Algebra 6.4 Solving Systems by Reducing Matrices Example 1 – Reduced Matrices For each of the following matrices, determine whether it is reduced or not reduced. Solution: a. Not reduced b. Reduced c. Not reduced d. Reduced e. Not reduced f. Reduced

By using matrix reduction, solve the system Chapter 6: Matrix Algebra 6.4 Solving Systems by Reducing Matrices Example 3 – Solving a System by Reduction By using matrix reduction, solve the system Solution: Reducing the augmented coefficient matrix of the system, We have

Using matrix reduction, solve Chapter 6: Matrix Algebra 6.4 Solving Systems by Reducing Matrices Example 5 – Parametric Form of a Solution Using matrix reduction, solve Solution: Reducing the matrix of the system, We have and x4 takes on any real value.

6.5 Solving Systems by Reducing Matrices (continued) Chapter 6: Matrix Algebra 6.5 Solving Systems by Reducing Matrices (continued) Example 1 – Two-Parameter Family of Solutions Using matrix reduction, solve Solution: The matrix is reduced to The solution is

is called a homogeneous system if c1 = c2 = … = cm = 0. Chapter 6: Matrix Algebra 6.5 Solving Systems by Reducing Matrices (Continue) The system is 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 solutions k = n  unique solution

2 equations (k), homogeneous system, 3 unknowns (n). Chapter 6: Matrix Algebra 6.5 Solving Systems by Reducing Matrices (Continue) Example 3 – Number of Solutions of a Homogeneous System Determine 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.

Chapter 6: Matrix Algebra 6.6 Inverses Example 1 – Inverse of a Matrix When matrix CA = I, C is an inverse of A and A is invertible. Let and . Determine whether C is an inverse of A. Solution: Thus, matrix C is an inverse of A.

Method to Find the Inverse of a Matrix When matrix is reduced, , Chapter 6: Matrix Algebra 6.6 Inverses Example 3 – Determining the Invertibility of a Matrix Method to Find the Inverse of a Matrix When 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 have Matrix A is invertible where

Solve the system by finding the inverse of the coefficient matrix. Chapter 6: Matrix Algebra 6.6 Inverses Example 5 – Using the Inverse to Solve a System Solve the system by finding the inverse of the coefficient matrix. Solution: We have For inverse, The solution is given by X = A−1B:

6.7 Leontief’s Input-Output Analysis Chapter 6: Matrix Algebra 6.7 Leontief’s Input-Output Analysis Example 1 – Input-Output Analysis Entries 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.)

Divide entries by the total value of output to get A: Chapter 6: Matrix Algebra 6.7 Leontief’s Input-Output Analysis Example 1 – Input-Output Analysis Solution: Divide entries by the total value of output to get A: Final-demand matrix: Output matrix is