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Multiplying Matrices

Multiplying Matrices. 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. Objectives. Understand the properties of matrices with respect to multiplication.

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Multiplying Matrices

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  1. Multiplying Matrices

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

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

  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nand an npmatrix is an mpmatrix.

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

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

  8. Example 1A: Identifying Matrix Products Tell whether the product is defined. If so, give its dimensions. A3  4 and B4  2; AB A B AB 344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. Example 1B: Identifying Matrix Products Tell whether the product is defined. If so, give its dimensions. C1  4 and D3  4; CD C D 1434 The inner dimensions are not equal (4 ≠ 3), so the matrix product is not defined.

  10. Check It Out! Example 1a Tell whether the product is defined. If so, give its dimensions. P2  5 Q5  3 R4  3 S4  5 QP Q P 5325 The inner dimensions are not equal (3 ≠ 2), so the matrix product is not defined.

  11. Check It Out! Example 1b Tell whether the product is defined. If so, give its dimensions. P2  5 Q5  3 R4  3 S4  5 SR S R 4543 The inner dimensions are not equal (5 ≠ 4), so the matrix product is not defined.

  12. Check It Out! Example 1c Tell whether the product is defined. If so, give its dimensions. P2  5 Q5  3 R4  3 S4  5 SQ S Q 4553 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.

  14. 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.

  15. Example 2A Continued Multiply row 1 of W and column 1 of X as shown. Place the result in wx11. 3(4) + –2(5)

  16. Example 2A Continued Multiply row 1 of W and column 2 of X as shown. Place the result in wx12. 3(7) + –2(1)

  17. Example 2A Continued Multiply row 1 of W and column 3 of X as shown. Place the result in wx13. 3(–2) + –2(–1)

  18. Example 2A Continued Multiply row 2 of W and column 1 of X as shown. Place the result in wx21. 1(4) + 0(5)

  19. Example 2A Continued Multiply row 2 of W and column 2 of X as shown. Place the result in wx22. 1(7) + 0(1)

  20. Example 2A Continued Multiply row 2 of W and column 3 of X as shown. Place the result in wx23. 1(–2) + 0(–1)

  21. Example 2A Continued Multiply row 3 of W and column 1 of X as shown. Place the result in wx31. 2(4) + –1(5)

  22. Example 2A Continued Multiply row 3 of W and column 2 of X as shown. Place the result in wx32. 2(7) + –1(1)

  23. Example 2A Continued Multiply row 3 of W and column 3 of X as shown. Place the result in wx33. 2(–2) + –1(–1)

  24. 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.

  25. 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.

  26. 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.

  27. 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.

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

  29. Example 3: Inventory Application 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. Use a product matrix to find the sales of each store for each day.

  30. Fri Sat Sun Video World Star Movies Example 3 Continued On Saturday, Video World made $851.05 and Star Movies made $832.50.

  31. 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.

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

  33. 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.

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

  35. Example 4A: Finding Powers of Matrices Evaluate, if possible. P3

  36. Example 4A Continued

  37. Example 4A Continued Check Use a calculator.

  38. Example 4B: Finding Powers of Matrices Evaluate, if possible. Q2

  39. Check It Out! Example 4a Evaluate if possible. C2 The matrices cannot be multiplied.

  40. Check It Out! Example 4b Evaluate if possible. A3

  41. Check It Out! Example 4c Evaluate if possible. B3

  42. Check It Out! Example 4d Evaluate if possible. I4

  43. Lesson Quiz Evaluate if possible. 1.AB 2.BA 3.A2 4.BD 5.C3

  44. Lesson Quiz Evaluate if possible. 1.AB 2.BA 3.A2 4.BD 5.C3 not possible not possible

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