1. Chapter 6
Matrix Algebra

2. 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

3. 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

4. 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.

5. 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)

6. 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.

7. 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

8. 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

9. 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.

10. 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:

11. 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

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

13. 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.

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

15. 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

16. 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.

17. 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:

18. 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:

19. 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

20. 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

21. 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.

22. 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

23. 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

24. 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.

25. 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

26. 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

27. 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.

28. 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.

29. 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

30. 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:

31. 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.)

32. 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