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Activity 8.5.1 Solving A Recycling Matrix EquationHere is a quick review of solving a linear system with two variables with a matrix equation:Two basketball teams are going into the third period with Team A leading Team B 70 to 64. Team A has made as many three point shots as team B has made two point shots. Team B has made two fewer three point shots than Team A has made two point shots. They both have made ten free throws. How many of each shot did each team make?Assign variables: Let x be the number of three point shots made by team ALet y be the number of two point shots made by team A.Write the point equation for both teams:Write the matrix equation :Use the formula for the inverse matrix: 1detAd-b-ca to find the inverse matrix:Multiply the inverse matrix by 6060 to find the solution:Verify that your results match these:Team A scored 12 threes and 12 twos for a total score of 36 + 24 + 10 = 70Team B scored 10 threes and 12 twos for a total score of 30 + 24 + 10 = 64 The same procedure can be used to solve equations involving three variables. For example, your school’s reduce/reuse/recycle initiative not only includes textile collection, but encouraging families to improve recycling efforts with bottles, cans, and newsprint. This naturally leads to assembling and analyzing data which is often linear and often involves more than two variables. For example:x=number of aluminum cans collectedy=number of plastic bottles collectedz=pounds of newspaper collectedThe equation for the value in dollars of the collected items is: 0.05x+0.05y+0.5z (1)The equation for the weight in pounds of the collected items is: 0.04x+0.045y+z (2)The equation for the reduction in pounds of CO2 emissions is: 0.4x+0.0765y+2.5z (3)1a. Explain the meaning of the term 0.05x in equation (1). ______1b. Explain the meaning of the term 0.045y in equation (2). ___________1c. Explain the meaning of the term 2.5z in equation (3)._______________Suppose we wished to know what combination of aluminum cans, plastic bottles and newsprint collected would result in $1000 gained, 1000 pounds collected, and a 5000 pound reduction in carbon dioxide emissions into the atmosphere? This can be solved using the exact same strategy we used with two variables. Write the matrix equation using the coefficients for the three equations:The next step is to find the inverse of the matrix. We have not yet learned how find the inverse of a 3×3 matrix by hand, so find the inverse with your graphing calculator. Check to see that your result matches this matrix:3.86-9.302.9532.15-8.04-3.22-1.601.73.027Solve the system by multiplying by the inverse matrix using your graphing calculator:xyz=3.86-9.302.9532.15-8.04-3.22-1.601.73.027100010005000=So in order to have $1000 gained, 1000 pounds collected, and a 5000 pound reduction in CO2, we must collect how many bottles, cans and pounds of newspaper?As in this example, we frequently use technology to find the determinant and the inverse of a 3×3 matrix. There is the method for finding them by hand. The method uses minors, cofactors and determinants so we first need to learn how to find minors, cofactors and determinants. We use as an example the 3×3 matrix: A=123456789A minor for an element aij in a square matrix B is a real number that is the determinant of a smaller matrix within a 3×3 matrix formed by crossing out row i and column j. For example, to find the minor of a11, cross out the first row and the first column and find the determinant of the matrix formed by remaining entries. The determinant of a matrix is written with straight bars instead of brackets.100208737380997212984500123456789 The minor for element a11 is the determinant of matrix 5689 which is written as:5689=45-48=-33319407914910020879149123456789 The minor for element a12 is 4679=36-42=-65948737717410020865109123456789 The minor for element a13 is 4578=32-35=-3There are nine minors for a 3×3 matrix. Why? _________. Here are a few more examples:59487364335100208252225123456789 The minor for element a23 is 1278=8-14=-610020841355033194075348123456789 The minor for element a32 is 1436=6-12=-6Find the remaining four minors for matrix A. Minors are used to find determinants and also to find the inverse of a matrix. First let’s see how we use them to find determinants. First a minor must be multiplied by a factor or 1 or -1 to determine a cofactor. A cofactor can be defined as Cij = (-1)i+j(Mij) where Mij is the corresponding minor. We find the determinant of a square matrix by using the cofactors across any row or down any column. For example, to find the determinant of matrix A by going across the first row, we will need the cofactors for the three elements in the first row and will then find their sum.This is how it works: To find the determinant of 123456789 which we write as: 123456789 by cofactor expansion across the first row:a11cofactor for entry a11+a12cofactor for entry a12+a13 cofactor for entry a13=(1)+15689+(2)-14679+(3)+14578=145-48-236-42+332-35=-3+12-9=0 To find the determinant working down the third column:a13cofactor for a13+a23cofactor for a23+a33cofactor for a33=332-35-68-14+95-8=-9+36-27=0 Notice we got same result as expected. There are four other ways to compute the determinant of this 3×3 matrix using cofactor expansion. Compute the determinant 123456789 using the second row. Compute 123456789 using the third pute 123456789 using the first pute 123456789 using the second column. Did they all match? The determinant of this matrix is 0. It can be mathematically proven that only matrices with nonzero determinants have an inverse. Recall that we did this for 2 by 2 matrices in investigation four. This matrix is singular and does not have an inverse. ................
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