1. BUSINESS MATHEMATICS & STATISTICS

2. Lecture 8 Discount_interest 1

6. LECTURE 9 Review Lecture 8 Matrices Matrix Applications using Excel

7. Qestions Where can we use Matrices? Typical applications? What is a Matrix? What are Matrix operations? Excel Matrix Functions?

8. Where can we use Matrices? Many applications in business and industry Where large amounts of data are processed daily

9. Typical Applications Practical questions in modern business and economic management using econometrics Network Analysis Decision Networks Optimization (Linear Programming) Analysis of data Computer graphics

10. What is a Matrix? A Matrix is a rectangular array of numbers The plural of matrix is matrices Matrices are usually represented with capital letters Matrices A, B, C

11. Size YouthSMLXL Pants010344012 Shirts182529217 Shorts191348369 T-shirts277102414

12. Dimension Dimension or Order of a Matrix = Number of Rows x Number of Columns Example Matrix T has dimensions of 2x3 or the order of matrix T is 2x3

13. Row, column and Square Matrix Row Matrix dimensions 1xn Column Matrix matrix with dimensions nx1 Square Matrix matrix with dimensions nxn

14. Row Matrix Example: Matrix A = 1x4 Column Matrix Example: Matrix B = a 2x1 Square Matrix Example: Matrix C = a 3x3

15. Identity Matrix A square matrix with 1's on the main diagonal from the upper left to the lower right and 0's off the main diagonal. Denoted as I Subscript indicates the size of the identity matrix represents an identity matrix with dimensions nxn.

16. Multiplicative Identity With real numbers, the number 1 is referred to as a multiplicative identity Unique property: product a real number and 1 is that real number.1 is called a multiplicative identity For any real number n, 1x n = n and n x1 = n.

17. Multiplicative Identity With matrices, the identity matrix shares the same unique property as the number 1. A 2x2 identity matrix is a multiplicative inverse because for any 2x2 matrix A, I2xA = A and AxI2 = A

18. BUSINESS MATHEMATICS & STATISTICS

19. Example 1 An athletic clothing company manufactures T-shirts and sweat shirts in four differents sizes, small, medium, large, and x-large. The company supplies two major universities, the U of R and the U of S. The tables below show September's clothing order for each university

20. University of S's September Clothing Order S M L XL T-shirts 100 300 500 300 Sweat shirts 150 400 450 250 University of R's September Clothing Order S M L XL T-shirts 60 250 400 250 Sweat shirts 100 200 350 200

21. Matrix Representation The above information can be given by two matrices S and R as shown to the right

22. Matrix Operations Organize and interpret data using matrices Use matrices in business applications Add and subtract two matrices Multiply a matrix by a scalar Multiply two matrices Interpret the meaning of the elements within a product matrix

23. + = ADDITION

24. PRODUCTION The clothing company production in preparation for the universities' Septmber orders is shown by the table and corresponding matrix, P=

25. Over-Production - =

26. + = ADDITION

27. Addition and Subtraction of Matrices The sum or difference of two matrices is caluculated by adding or subtracting the corresponding elements of the matrices To add or subtract matrices, they must have the same dimensions.

28. POSSIBLE ? YES No

29. MULTIPLICATION 591mL 1L 2L Company A 20,000 5,500 10,600 Company B 18,250 7,000 11,000 Price 1.60 2.30 3.10 What is total revenue of Company A? What is total revenue of Company B?

30. MULTIPLICATION S= P= R= 20,000x1.6+5,500x2.3+10,600x3.1= 77,510 18,250x1.6+7,000x2.3+11,000x3.1=79,400

31. Matrix Functions in Microsoft Excel MDETERM Returns the matrix determinant of an array MINVERSE Returns the matrix inverse of an array MMULT Returns the matrix product of two arrays

32. MULTIPLICATION 591mL 1L 2L Company A 20,000 5,500 10,600 Company B 18,250 7,000 11,000 Price 1.60 2.30 3.10 What is total revenue of Company A? What is total revenue of Company B?

33. MULTIPLICATION S= P= R= 20,000x1.6+5,500x2.3+10,600x3.1= 77,510 18,250x1.6+7,000x2.3+11,000x3.1=79,400