The prices for different sandwiches are presented at right.

1. The table shows the top scores for girls in barrel racing at the 2004 National High School Rodeo finals. The data can be presented in a table or a spreadsheet as rows and columns of numbers.

2. You can also use a matrix to show table data. A matrix is a rectangular array of numbers enclosed in brackets. Matrix A has two rows and three columns. A matrix with m rows and n columns has dimensionsm n, read “m by n,” and is called an mn matrix. Matrix A has dimensions 2  3. Each value in a matrix is called an element of the matrix.

3. We name an element by its location in a matrix, where the number of the Row comes first, and then the number of the Column. This can also be expressed by using the lower case matrix letter with row and column number as subscripts. The score 16.206 is located in row 2 column 1, so the name of this element is a2,1 (or Row 2, Column 1), and it’s value is 16.206.

4. 3.95 5.95 3.75 5.60 3.50 5.25 P = Example 1: Displaying Data in Matrix Form The prices for different sandwiches are presented at right. A. Display the data in matrix form. B. What are the dimensions of P? P has three rows and two columns, so it is a 3  2 matrix.

5. Example 2: Displaying Data in Matrix Form The prices for different sandwiches are presented at right. C. What is element P3,2? What does is represent? The entry at P3,2, in row 3 column 2, is 5.25. It is the price of a 9 in. tuna sandwich. D. What is the name of the element 5.95? The element 5.95 is P1,2. It is Row 1 Column 2

6. Check It Out! Example 1 Use matrix M to answer the questions below. a. What are the dimensions of M? 3  4 b. What is the value of m3,2? 11 c. The entry 0 appears for which two elements? m1,4 and m2,3

7. Example 3: Point Matrices Points in the coordinate plane can also be represented by a matrix. This is called aPoint Matrix. Row 1 is always representing the x-coordinate, and Row 2 is always representing the y-coordinate. So any point matrix will only ever have 2 rows, since coordinates only have two values. x y (x,y) =

8. Example 3a: Point Matrices Triangle ABC is graphed in the coordinate plane on the left. Write its vertices as a point matrix. -The x-coordinates become the elements in the first row of the matrix. A = (-1, 1) B = (-1, 3) C = (1, 2) -The y-coordinates become the elements in the 2nd row of the matrix. A B C X Y -1 -1 1 1 3 2

9. Example 3b: Point Matrices Triangle A’B’C’ is graphed in the coordinate plane on the left. Write its vertices as a point matrix. -Put the x-coordinates as the elements in the first row of the matrix. A’ = (-1, -1) B’ = (-3, -1) C’ = (-2, 1) -Put the y-coordinates as the elements in the 2nd row of the matrix. A’ B’ C’ X Y -1 -3 -2 -1 -1 1

10. You can add or subtract two matrices only if they have the same dimensions. If the dimensions are the same, then you can add the corresponding elements: Those that are in the same location in each matrix (those with the same name)

11. 3 –2 1 0 3 + 1–2 + 4 1 + (–2)0 + 3 4 2 –1 3 + = = 1 4 –2 3 Example 2A: Finding Matrix Sums and Differences Add or subtract, if possible. 3 –2 1 0 4 7 2 5 1 –1 1 4 –2 3 2 –2 3 1 0 4 W = , X = , Y = , Z = W + Y Add each corresponding entry. W + Y =

12. 2 –2 3 1 0 4 4 7 2 5 1 –1 2 9 –1 4 1 –5 – = Example 2B: Finding Matrix Sums and Differences Add or subtract, if possible. 3 –2 1 0 4 7 2 5 1 –1 1 4 –2 3 2 –2 3 1 0 4 W = , X = , Y = , Z = X – Z Subtract each corresponding element. X – Z =

13. Example 2C: Finding Matrix Sums and Differences Add or subtract, if possible. 3 –2 1 0 4 7 2 5 1 –1 1 4 –2 3 2 –2 3 1 0 4 W = , X = , Y = , Z = X + Y X is a 2  3 matrix, and Y is a 2  2 matrix. Because X and Y do not have the same dimensions, they cannot be added.

14. 4 0 –8 6 2 18 4 –1 –5 3 2 8 0 1 –3 3 0 10 4 + 0 –1 + 1 –5 + (–3) 3 + 3 2 + 0 8 + 10 + = Check It Out! Example 2A Add or subtract if possible. 4 –2 –3 10 2 6 3 2 0 –9 –5 14 4 –1 –5 3 2 8 0 1 –3 3 0 10 A = , C = , D = B = , B + D Add each corresponding entry. B + D =

15. Check It Out! Example 2B Add or subtract if possible. 4 –2 –3 10 2 6 3 2 0 –9 –5 14 4 –1 –5 3 2 8 0 1 –3 3 0 10 A = , C = , D = B = , B – A B is a 2  3 matrix, and A is a 3  2 matrix. Because B and A do not have the same dimensions, they cannot be subtracted.

16. 0 1 –3 3 0 10 4 –1 –5 3 2 8 –4 2 2 0 –2 2 – = Check It Out! Example 2C Add or subtract if possible. 4 –2 –3 10 2 6 3 2 0 –9 –5 14 4 –1 –5 3 2 8 0 1 –3 3 0 10 A = , C = , D = B = , D – B Subtract corresponding entries. D – B =

17. 0 1 –3 3 0 10 4 –1 –5 3 2 8 4 0 -8 6 2 18 + = Commutative Property We already found B+D, Now find D+B 4 –2 –3 10 2 6 3 2 0 –9 –5 14 4 –1 –5 3 2 8 0 1 –3 3 0 10 A = , C = , D = B = , D+B = Are the results the same? Yes! So Matrix Addition is Commutative

18. 4 -1 –5 3 2 8 0 1 3 3 0 10 4 -2 -2 0 2 -2 – = Commutative Property We already found D-B, NOW find B-D: 4 –2 –3 10 2 6 3 2 0 –9 –5 14 4 –1 –5 3 2 8 0 1 –3 3 0 10 A = , C = , D = B = , B-D Subtract corresponding entries. B-D = Are the results the same? NO! So, Matrix Subtraction is NOT Commutative

19. Associative Property Consider Three Matrices: A,B, and C. Assuming that they all have the same dimensions, Would changing the grouping affect the result of Matrix Addition? Does A+(B+C) = (A+B)+C ???? YES! So, Matrix Addition is also Associative Would changing the grouping affect the result of Matrix Subtraction? Does A-(B-C) = (A-B)-C ???? NO! So, Matrix Subtraction is also NOT Associative

20. You can multiply a matrix by a number, called a scalar. To find the product of a scalar and a matrix, or the scalar product, multiply each element by the scalar.

21. 30 17.5 25 14 40 22.5 150 87.5 125 70 200 112.5 150 87.5 125 70 200 112.5 150 87.5 125 70 200 112.5 – 120 70 100 56 160 90 Check It Out! Example 4 Use a scalar product to find the prices if a 20% discount is applied to the ticket service prices. You can multiply by 0.2 and subtract from the original numbers. – 0.2 =

22. Example 4b: Simplifying Matrix Expressions 3 –2 1 0 2 –1 1 4 –2 3 0 4 4 7 2 5 1 –1 P = R = Q= Evaluate 3P — Q, if possible. P and Q do not have the same dimensions; they cannot be subtracted after the scalar products are found.

23. 3 12 –6 9 0 12 1 4 –2 3 0 4 3 –2 1 0 2 –1 3 –2 1 0 2 –1 3(1) 3(4) 3(–2) 3(3) 3(0) 3(4) 3 –2 1 0 2 –1 = = 3 – – – 0 14 –7 9 –2 13 Example 4c: Simplifying Matrix Expressions 3 –2 1 0 2 –1 1 4 –2 3 0 4 4 7 2 5 1 –1 P = R = Q= Evaluate 3R — P, if possible. =

24. Check It Out! Example 4d 4 –2 –3 10 4 –1 –5 3 2 8 3 2 0 –9 D = [6 –3 8] A = B = C = Evaluate 3B + 2C, if possible. B and C do not have the same dimensions; they cannot be added after the scalar products are found.

25. 4 –2 –3 10 3 2 0 –9 = 2 –3 8 –4 –6 20 9 6 0 -27 –1 –10 –6 47 2(4) 2(–2) 2(–3) 2(10) 3(3) 3(2) 3(0) 3(–9) = - = = - Check It Out! Example 4e 4 –2 –3 10 4 –1 –5 3 2 8 3 2 0 –9 D = [6 –3 8] A = B = C = Evaluate 2A – 3C, if possible.

26. Check It Out! Example 4f 4 –2 –3 10 4 –1 –5 3 2 8 3 2 0 –9 D = [6 –3 8] A = B = C = Evaluate D + 0.5D, if possible. = [6 –3 8] + 0.5[6 –3 8] = [6 –3 8] + [0.5(6) 0.5(–3) 0.5(8)] = [6 –3 8] + [3 –1.5 4] = [9 –4.5 12]

27. Lesson Quiz 1. What are the dimensions of A? 2. What is entry D1,2? Evaluate if possible. 3.10(2B + D)4.C + 2D 5. Graph D in the coordinate plane as ΔABC 3  2 –2 Not possible C. A. B.