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

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

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

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  1. Objectives Understand the properties of matrices with respect to multiplication. Multiply two 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. • Matrices A and B can be multiplied only if the number of columns in A equalsthe number of rows in B. • The product of an mnand an npmatrix is an mpmatrix.

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

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

  4. 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.

  5. Find the product, if possible. WX Check the dimensions. W is 3  2 , X is 2  3 . WX is defined and is 3  3. Multiply row 1 of W and column 1 of X as shown. Place the result in wx11. 3(4) + –2(5)

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

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

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

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

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

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

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

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

  14. 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.

  15. 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.

  16. 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.

  17. 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.

  18. 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.

  19. Fri Sat Sun Video World Star Movies On Saturday, Video World made $851.05 and Star Movies made $832.50.

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

  21. Evaluate, if possible. P3

  22. Check Use a calculator. Evaluate, if possible. Q2

  23. Check Use a calculator. Evaluate if possible. A3 C2 The matrices cannot be multiplied.

  24. Evaluate if possible. B3 I4

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