2. What is a Matrices? A matrix is a rectangular array of data entries (elements) displayed in rows and columns and enclosed in brackets. The number of rows and columns in the matrix determines its dimensions.

3. The number of rows is always reported before the number of columns. Rows go across Columns go up and down Example: [ ] 2 4 6 9 5 3 This is a 3 x 2 matrix Because you have 3 rows and 2 columns

4. Name These Matrices! [] 4 5 6 2 3 4 [ [ ] ] 16 -4 12 6 12 –3 8 4 3 4 8 8 2 4 4 6

5. Name These Matrices! [] 4 5 6 2 3 4 [ [ ] ] 16 -4 12 6 12 –3 8 4 3 4 8 8 2 4 4 6 2 x 3 matrix 2 x 3 matrix 3 x 4 3 x 4matrix 2 x 2 2 x 2matrix

6. ADDING MATRICES To add 2 or more matrices, we must be sure that we have the same frame. That is, the same number of rows and columns. If you don’t have the same dimensions, then the question can not be added.

7. ADDING MATRICES When adding, you add the 2 numbers that are in the exact same position in each matrix. These two numbers are called “corresponding entries”

8. EXAMPLE [ ][[ ] ] 5.37.4 + 8.59.1 = 5.3 + 8.5 (5.3 + 8.5) 7.4 + 9.1 (7.4 + 9.1) = [ ] 13.816.5 2 by 1 matrix

9. EXAMPLE #2 [ ][[ ] ] 2 7 3 5 9 4 + 5 6 4 2 3 2 = 2+5) (7+6) (3+4) (2+5) (7+6) (3+4) (5+2) (9+3) (4+2) = [ ] 7 13 7 7 12 6 2 by 3 matrix

10. YOU TRY [ ][[ ] ] 3 7 5 24 6 1 1 1 + 5 6 2 64 1 1 3 6 = = [ ] __ by __ matrix

11. SOLUTION [ ][[ ] ] 3 7 5 2 4 6 1 1 1 + 5 6 2 6 4 1 1 3 6 = = [ ] 3 by 3 matrix 3+5) (7+6) (5+2) (3+5) (7+6) (5+2) (2+6) (4+4) (6+1) (1+1) (1+3) (1+6) 8 13 7 8 8 7 2 4 7

12. SUBTRACTION The same rules apply as for addition [ ][[ ] ] 5 8 3 6 - 25 1 4 = = [ ] 2 by 2 matrix (5-2) (8-5) (3-1) (6-4) 33 2 2

13. SUBTRACTION The same rules apply as for addition [ ][[ ] ] 8 4 7 9 - -2 3 5 -11 = = [ ] 2 by 2 matrix (8-(-2)) (4-3) (7-5) (9-(-11)) 10 1 2 20

14. YOU TRY [ ][[ ] ] - = = [ ] __ by __ matrix 5 -7 9 2 4 -6 8 -7 5 2 -3 5 6 -3 7 -5 9 1

15. SOLUTION [ ][[ ] ] - = = [ ] 3 by 3 matrix 5 -7 9 2 4 -6 8 -7 5 2 -3 5 6 -3 7 -5 9 1 5-2) (-7-(-3)) (9-5) (5-2) (-7-(-3)) (9-5) (2-6) (4-(-3)) (-6-7) (8- (-5)) (-7-9) (5-1) 3 -4 4 -4 7 -13 13 -16 4

16. Multiplying Matrices Example: 2 [ ] = [ ] 3 5 4 6 6 10 14 12 Multiply each term by the number outside the matrix, in this case you are doubling the matrix

17. You try 3 [ ]= - 3 -2 7 4 5 -2 2 -30 6

18. Matrices Word Problems 1. Long-distance rates for calls within Canada for 4 telephone companies are given in the first table on the next slide: Due to competition, the companies adjusted their rates as shown in the second table: a. Display the long-distance rates in matrix A and the adjustments to the rates in matrix B. b. Determine A + B. Explain what the resulting matrix C represents.

19. Rates per Minute From 8 am to 6 pm Rates per Minute From 6 pm to 8 am Company 1$0.15$0.10 Company 2$0.14 Company 3$0.18$0.08 Company 4$0.14$0.12 Adjustment to Daytime Rates ( per minute) Adjustment to Evening Rates ( per minute) Company 1-$0.01$0.01 Company 2No change Company 3-$0.02$0.01 Company 4-$0.01

20. CLASS WORK Do Lesson 31 worksheet Remember to start reviewing for your exam.