1. MTH108 Business Math I
Lecture 20

2. Chapter 9
Matrix Algebra

3. Objectives
Provide an understanding of the nature of the matrix and matrix representation of data
Provide an understanding of the algebra of matrices
Provide applications of matrices and matrix algebra

4. Main Topics Introduction to matrices Special types of matrices
Matrix operations
The determinant
The inverse of a matrix
Some applications

5. Review Matrix: need; definition
Basic facts: notation; elements; dimension; examples
Types of matrices: row vector; column vector; square matrix; identity matrix; diagonal matrix; zero matrix
Transpose of a matrix
Matrix operations: addition; subtraction; scalar multiplication
Properties of matrix operations

6. Today’s Topics Matrix operations; matrix multiplication
Representation of system of equations

7. Matrix multiplication
Assume that a matrix A having dimension is to be multiplied with matrix having dimension . The product C = AB is to be defined a matrix of order whose entry in ith row and jth column is obtained as: Sum the products formed by multiplying in order each entry in row I of A by the corresponding entry in column j of B.

8. Properties
The matrix product AB is defined when rows of A is equal to the columns of B.
The resulting product will have the order of row of A and column of B.
The order of multiplication matters.

9. Examples
1) 2)

10. Examples 3)

11. Examples 3)

12. Examples
4) 5)

13. Examples 6)

14. Examples
7) Cost Vector Suppose that the prices for products A, B and C are given by the price vector If quantities of A, B and C that are purchased are given by the column vector

15. Examples
7) Cost Vector Then total cost C is given by:

16. Properties of matrix multiplication

18. Equality of matrices
Definition: Two matrices A and B are equal if and only if, they have the same dimension and e.g.

19. A matrix equation can define a system of equations. E.g.

20. Solve the equation

21. Representation of an equation
An equation may be represented using the matrix form. E.g.

22. Representation of an equation
A linear equation of the form

23. Representation of system of equations
A system of equations of the form

24. Consider the system of equations

25. The system of equations

26. Summary Matrix multiplication Properties Equality of matrices
Matrix equation
System of equations
Section 9.3 Follow up exercises