2. Warm Up State the dimensions of each matrix. 1. [3 1 4 6] 2. Calculate. 3. 3(–4) + ( – 2)(5) + 4(7) 4. ( – 3)3 + 2(5) + ( – 1)(12) 1  4 3  2 6 –11

3. Understand the properties of matrices with respect to multiplication. Multiply two matrices. Objectives

4. matrix product square matrix main diagonal multiplicative identity matrix Vocabulary

5. In Lesson 4-1, you multiplied matrices by a number called a scalar. You can also multiply matrices together. The product of two or more matrices is the matrix product. The following rules apply when multiplying matrices. Matrices A and B can be multiplied only if the number of columns in A equals the number of rows in B. The product of an m  n and an n  p matrix is an m  p matrix.

6. An m  n matrix A can be identified by using the notation A m  n.

7. The CAR key: Columns (of A) As Rows (of B) or matrix product AB won’t even start Helpful Hint

8. Tell whether the product is defined. If so, give its dimensions. Example 1A: Identifying Matrix Products A 3  4 and B 4  2 ; AB A BAB 3  4 4  2 = 3  2 matrix The inner dimensions are equal (4 = 4), so the matrix product is defined. The dimensions of the product are the outer numbers, 3  2.

9. Tell whether the product is defined. If so, give its dimensions. Example 1B: Identifying Matrix Products C 1  4 and D 3  4 ; CD C D 1  4 3  4 The inner dimensions are not equal (4 ≠ 3), so the matrix product is not defined.

10. Tell whether the product is defined. If so, give its dimensions. P 2  5 Q 5  3 R 4  3 S 4  5 Q P 5  3 2  5 The inner dimensions are not equal (3 ≠ 2), so the matrix product is not defined. Check It Out! Example 1a QP

11. Tell whether the product is defined. If so, give its dimensions. P 2  5 Q 5  3 R 4  3 S 4  5 S R 4  5 4  3 Check It Out! Example 1b SR The inner dimensions are not equal (5 ≠ 4), so the matrix product is not defined.

12. Tell whether the product is defined. If so, give its dimensions. P 2  5 Q 5  3 R 4  3 S 4  5 S Q 4  5 5  3 Check It Out! Example 1c SQ The inner dimensions are equal (5 = 5), so the matrix product is defined. The dimensions of the product are the outer numbers, 4  3.

13. Just as you look across the columns of A and down the rows of B to see if a product AB exists, you do the same to find the entries in a matrix product.

15. Example 2A: Finding the Matrix Product Find the product, if possible. WX Check the dimensions. W is 3  2, X is 2  3. WX is defined and is 3  3.

16. Example 2A Continued Multiply row 1 of W and column 1 of X as shown. Place the result in wx 11. 3(4) + –2(5)

17. Example 2A Continued Multiply row 1 of W and column 2 of X as shown. Place the result in wx 12. 3(7) + –2(1)

18. Example 2A Continued Multiply row 1 of W and column 3 of X as shown. Place the result in wx 13. 3(–2) + –2(–1)

19. Example 2A Continued Multiply row 2 of W and column 1 of X as shown. Place the result in wx 21. 1(4) + 0(5)

20. Example 2A Continued Multiply row 2 of W and column 2 of X as shown. Place the result in wx 22. 1(7) + 0(1)

21. Example 2A Continued Multiply row 2 of W and column 3 of X as shown. Place the result in wx 23. 1(–2) + 0(–1)

22. Example 2A Continued Multiply row 3 of W and column 1 of X as shown. Place the result in wx 31. 2(4) + –1(5)

23. Example 2A Continued Multiply row 3 of W and column 2 of X as shown. Place the result in wx 32. 2(7) + –1(1)

24. Example 2A Continued Multiply row 3 of W and column 3 of X as shown. Place the result in wx 33. 2(–2) + –1(–1)

25. Example 2B: Finding the Matrix Product Find each product, if possible. XW Check the dimensions. X is 2  3, and W is 3  2 so the product is defined and is 2  2.

26. Example 2C: Finding the Matrix Product Find each product, if possible. XY Check the dimensions. X is 2  3, and Y is 2  2. The product is not defined. The matrices cannot be multiplied in this order.

27. Check It Out! Example 2a Find the product, if possible. BC Check the dimensions. B is 3  2, and C is 2  2 so the product is defined and is 3  2.

28. Check It Out! Example 2b Find the product, if possible. CA Check the dimensions. C is 2  2, and A is 2  3 so the product is defined and is 2  3.

29. Businesses can use matrix multiplication to find total revenues, costs, and profits.

30. Two stores held sales on their videos and DVDs, with prices as shown. Use the sales data to determine how much money each store brought in from the sale on Saturday. Example 3: Inventory Application Use a product matrix to find the sales of each store for each day.

31. Example 3 Continued On Saturday, Video World made $851.05 and Star Movies made $832.50. Fri SatSun Video World Star Movies

32. Check It Out! Example 3 Change Store 2’s inventory to 6 complete and 9 super complete. Update the product matrix, and find the profit for Store 2. Skateboard Kit Inventory Complete Super Complete Store 11410 Store 269

33. Check It Out! Example 3 Use a product matrix to find the revenue, cost, and profit for each store. Revenue Cost Profit Store 1 Store 2 The profit for Store 2 was $819.

34. A square matrix is any matrix that has the same number of rows as columns; it is an n × n matrix. The main diagonal of a square matrix is the diagonal from the upper left corner to the lower right corner. The multiplicative identity matrix is any square matrix, named with the letter I, that has all of the entries along the main diagonal equal to 1 and all of the other entries equal to 0.

35. Because square matrices can be multiplied by themselves any number of times, you can find powers of square matrices.

36. Example 4A: Finding Powers of Matrices Evaluate, if possible. P 3

37. Example 4A Continued

38. Check Use a calculator.

39. Example 4B: Finding Powers of Matrices Evaluate, if possible. Q 2

40. Check It Out! Example 4a C2C2 Evaluate if possible. The matrices cannot be multiplied.

41. Check It Out! Example 4b A3A3 Evaluate if possible.

42. Check It Out! Example 4c B3B3 Evaluate if possible.

43. Check It Out! Example 4d I4I4 Evaluate if possible.

44. Lesson Quiz Evaluate if possible. 1. AB 2. BA 3. A 2 4. BD 5. C 3

45. Lesson Quiz Evaluate if possible. 1. AB 2. BA 3. A 2 4. BD 5. C 3 not possible