1. BUSINESS MATHEMATICS
&
STATISTICS

2. LECTURE 10
Review Lecture 9
Matrices

3. 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

4. University of S's September Clothing Order S M L XL
T-shirts
Sweat shirts  
University of R's September Clothing Order
T-shirts
Sweat shirts

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

6. 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

7. ADDITION + =

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

9. Over-Production - =

10. ADDITION + =

11. 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.

12. POSSIBLE ?
YES
YES No

13. MULTIPLICATION 591mL 1L 2L Company A 20,000 5,500 10,600
Company B 18, , ,000
Price
What is total revenue of Company A?
What is total revenue of Company B?

14. MULTIPLICATION S= P= R= 20,000x1.6+5,500x2.3+10,600x3.1= 77,510

15. MULTIPLICATION RULES If matrix A = a mxn matrix
matrix B = nxp matrix, then
AB = mxp matrix
Entry of AB in i-th row and j-th column =
i-th row of matrix A x j-th row of matrix B
Multiply corresponding elements (first by first, second by second, and so on) Number of columns of A must equal the number of rows of B

16. Number of rows of A must be equal to
Dimensions of Dimensions the Product Matrix
Matrices
A = 3x3      B = 3x AB = 3x2
B = 3x A = 3x No product
Example Calculation:
12x21+13x23+14x26
In the second case
21x12+22x15+?x18
No term for multiplication.
Number of rows of A must be equal to
columns of B

17. Multiplicative inverse of a matrix
MATRIX INVERSION
for any 2x 2matrix A,
I2x A = A and AxI2= A
Anx1/An = In & Anx1/An = In
Multiplicative inverse of a matrix =
x = = x

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

19. MINVERSE MINVERSE(array)
Array   is a numeric array with an equal number of rows and columns
Inverse of the matrix A1:B2.
Column A Column B
Row d/(a*d-b*c) b/(b*c-a*d)
Row c/(b*c-a*d) a/(a*d-b*c)
Array formula F2
Enter formula
Ctrl + Shift + Enter

20. MINVERSE

22. Array is a numeric array with an equal number of rows and columns
MDETERM
MDETERM(array)
Array   is a numeric array with an equal number of rows and columns
Array, A1:C3, the determinant is defined as: MDETERM(A1:C3) equals A1*(B2*C3-B3*C2) + A2*(B3*C1-B1*C3) + A3*(B1*C2-B2*C1)

24. Array1, array2 are the arrays you want to multiply
MMULT
Returns the matrix product of two arrays. The result is an array with the same number of rows as array1 and the same number of columns as array2.
MMULT(array1,array2)
Array1, array2   are the arrays you want to multiply
Array Formula

26. BUSINESS MATHEMATICS
&
STATISTICS