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EXAMPLE 4

2 –3. 1 0. The matrices are both 2 2 , so their product is defined. Use the following steps to find the elements of the product matrix. –1 8. 4 5. EXAMPLE 4. Multiply matrices. Multiply. SOLUTION. 1 0. 1 ( 2 ) + 0 ( –1 ) ?. 2 –3. =.

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EXAMPLE 4

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  1. 2 –3 1 0 The matrices are both 2 2, so their product is defined. Use the following steps to find the elements of the product matrix. . –1 8 4 5 EXAMPLE 4 Multiply matrices Multiply SOLUTION

  2. 1 0 1(2) + 0(–1) ? 2 –3 = ? ? 4 5 –18 EXAMPLE 4 Multiply matrices STEP 1 Multiply the numbers in the first row of the first matrix by the numbers in the first column of the second matrix. Put the result in the first row, first column of the product matrix.

  3. 1 0 1(2) + 0(–1) 1(–3) + 0(8) 2 –3 = ? ? 4 5 –1 8 EXAMPLE 4 Multiply matrices STEP 2 Multiply the numbers in the first row of the first matrix by the numbers in the second column of the second matrix. Put the result in the first row, second column of the product matrix.

  4. 1 0 1(2) + 0(–1) 1(–3) + 0(8) 2 –3 = 4(2) + 5(–1) ? 4 5 –18 EXAMPLE 4 Multiply matrices STEP 3 Multiply the numbers in the second row of the first matrix by the numbers in the first column of the second matrix. Put the result in the second row, first column of the product matrix.

  5. 1 0 1(2) + 0(–1) 1(–3) + 0(8) 2 –3 = 4(2) + 5(–1)4(–3) + 5(8) 4 5 –1 8 EXAMPLE 4 Multiply matrices STEP 4 Multiply the numbers in the second row of the first matrix by the numbers in the second column of the second matrix. Put the result in the second row, second column of the product matrix.

  6. 1(2) + 0(–1) 1(–3) + 0(8) 2 –3 = 3 28 4(2) + 5(–1) 4(–3) + 5(8) EXAMPLE 4 Multiply matrices STEP 5 Simplify the product matrix.

  7. Two softball teams submit equipment lists for the season. A bat costs $20, a ball costs $5, and a uniform costs $40. Use matrix multiplication to find the total cost of equipment for each team. EXAMPLE 5 Solve a real-world problem SOFTBALL

  8. EXAMPLE 5 Solve a real-world problem SOLUTION First, write the equipment lists and the costs per item in matrix form. You will use matrix multiplication, so you need to set up the matrices so that the number of columns of the equipment matrix matches the number of rows of the cost per item matrix.

  9. Dollars Bats Balls Uniforms Dollars 20 Bats ? 13 42 16 Women Women = 5 Balls 15 45 18 ? Men Men 40 Uniforms EXAMPLE 5 Solve a real-world problem = EQUIPMENT COST TOTAL COST

  10. 20 13(20) + 42(5) + 16(40) 1110 13 42 16 = = 5 15 45 18 1245 15(20) + 45(5) + 18(40) 40 ANSWER The total cost of equipment for the women’s team is $1110, and the total cost for the men’s team is $1245. EXAMPLE 5 Solve a real-world problem You can find the total cost of equipment for each team by multiplying the equipment matrix by the cost per item matrix. The equipment matrix is 2 3 and the cost per item matrix is 3 1, so their product is a 2 1 matrix.

  11. –3 B = [2 1] A = ANSWER 4 Yes; the number of columns in A is equal to the number of rows in B. 6.7 0 = C –9.3 5.2 for Examples 4 and 5 GUIDED PRACTICE Use the matrices below. Is the product defined? Explain. 6. AB

  12. –3 B = [2 1] A = 4 ANSWER 6.7 0 Yes; the number of columns in B is equal to the number of rows in A. = C –9.3 5.2 for Examples 4 and 5 GUIDED PRACTICE Use the matrices below. Is the product defined? Explain. 7. BA

  13. –3 B = [2 1] A = ANSWER 4 No; the number of columns in A is not equal to the number of rows in C. 6.7 0 = C –9.3 5.2 for Examples 4 and 5 GUIDED PRACTICE Use the matrices below. Is the product defined? Explain. 8. AC

  14. 3 8 = –4 7 1(3) + 0(–4) 1(8) + 0(7) 3 8 1 0 = 0(3) + 1(–4) 0(8) + 1(7) –4 7 0 1 for Examples 4 and 5 GUIDED PRACTICE Multiply. 9. 3 8 1 0 –4 7 0 1 SOLUTION

  15. 10. –3 [5 1] –2 –3 [5 1] = [5(–3) + 1(–2)] –2 for Examples 4 and 5 GUIDED PRACTICE SOLUTION = [ – 17 ]

  16. 15 –19 = –3 –5 11. 2 –4 5 1 5 1 1 –1 2 –4 5(2) + 1(5) 5(–4) + 1(1) 5 1 = 5 1 1 –1 1(2) –1(5) 1(–4) + –1(1) for Examples 4 and 5 GUIDED PRACTICE SOLUTION

  17. for Examples 4 and 5 GUIDED PRACTICE 12. WHAT IF? In Example 5, find the total cost for each team if a bat costs $25, a ball costs $4, and a uniform costs $35. SOLUTION First, write the equipment lists and the costs per item in matrix form. You will use matrix multiplication, so you need to set up the matrices so that the number of columns of the equipment matrix matches the number of rows of the cost per item matrix.

  18. Dollars Bats Balls Uniforms Dollars 25 Bats ? 13 42 16 Women Women = 4 Balls 15 45 18 ? Men Men 35 Uniforms for Examples 4 and 5 GUIDED PRACTICE = EQUIPMENT COST TOTAL COST

  19. 25 13(25) + 42(4) + 16(35) 1053 13 42 16 = = 4 15 45 18 1185 15(25) + 45(4) + 18(35) 35 ANSWER The total cost of equipment for the women’s team is $1053, and the total cost for the men’s team is $1185. for Examples 4 and 5 GUIDED PRACTICE You can find the total cost of equipment for each team by multiplying the equipment matrix by the cost per item matrix. The equipment matrix is 2 3 and the cost per item matrix is 3 1, so their product is a 2 1 matrix.

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