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Lesson 4–2 Objectives

Lesson 4–2 Objectives. Be able to multiply matrices. 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.

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Lesson 4–2 Objectives

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  1. Lesson 4–2 Objectives • Be able to multiply matrices

  2. 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.

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

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

  5. Example 1: 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.

  6. Example 1: Identifying Matrix Products Tell whether the product is defined. If so, give its dimensions. A3  4 and B4  2; BA B A BA 4324 = can’t be done

  7. Example 1: Identifying Matrix Products Tell whether the product is defined. If so, give its dimensions. C1  4 and D3  4; CD C D CD 1434 = can’t be done

  8. Example 1: Identifying Matrix Products Tell whether the product is defined. If so, give its dimensions. C1  4 and D3  4; DC D C DC 3414 = can’t be done

  9. 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.

  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 SR S R 4543 The inner dimensions are not equal (5 ≠ 4), so the matrix product is not defined.

  11. Check It Out! Example 1a 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.

  12. Example 2: 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.

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

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

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

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

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

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

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

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

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

  22. Example 2: Finding the Matrix Product Find the 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.

  23. Example 2: Finding the Matrix Product Find the 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.

  24. 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.

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

  26. 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.

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

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

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

  30. Check It Out! Example 4a Evaluate if possible. A3

  31. Check It Out! Example 4a Evaluate if possible. B3

  32. Check It Out! Example 4a Evaluate if possible. I4

  33. Lesson Assignment

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