LAB 11: MATRICES, SYSTEMS OF EQUATIONS and POLYNOMIAL MODELING

Transcription

1 LAB 11: MATRICS, SYSTMS OF QUATIONS and POLYNOMIAL MODLING Objectives: 1. Solve systems of linear equations using augmented matrices. 2. Solve systems of linear equations using matrix equations and inverse matrices. 3. Create polynomial models for given sets of data. Reference Topics: Matrices and Linear quations Operations with Matrices The Inverse of a Square Matrix Discussion In this lab, you will use the computer or calculator to help you solve systems of linear equations using matrices. You will then use these methods to find polynomial functions that fit given data. Finally, you will estimate the area of a given region with a curved boundary, by first finding an equation that represents the boundary, and then using linear approximations to that curve to estimate the area using trapezoids. In this lab, Part I, pp. 1-4, is for DRIV users only. Part 2, pp. 5-7, is for TI- 82 users only. The remainder of the lab is for both DRIV and the TI-82. Part 1 - Matrix Solutions on DRIV A. Solving Linear Systems using Augmented Matrices Consider the system of equations: x + 2y - 3z = 5 2x - 3y + 4z = 10 3x + 5y - 6z = Write the augmented matrix for this system in the space below. 2. nter the augmented matrix into the computer using the Declare Matrix commands, with 3 rows and 4 columns. Make sure the matrix is highlighted. In the Author command: 1. Type: row_reduce +You must use the underline symbol (SHIFT - ) 2. Press the F3 key and then press NTR You should see ROW_RDUC Now use the Simplify command to make DRIV row reduce the matrix. The result is a matrix in reduced row-echelon form. Write the result below.

2 5. Use this matrix to find the solutions to the original system. Write the results in the space below. 6. For each of the following systems, write down the augmented matrix. Then use the computer to solve the system using the Row_reduce statement. Write the resulting matrix. Then interpret this matrix to find the solutions to the original system. a. 2w + x - y + 2z = -16 3w + 4x + z = 1 w + 5x + 2y + 6z = -3 5w + 2x - y - z = 3 b. x + 2y + 3z = 4 5x + 6y + 7z = 8 9x + 10y + 11z = 12 B. Solving Linear Systems using Matrix quations and Inverse Matrices Consider the system of equations: x + 2y - 3z = 5 2x - 3y + 4z = 10 3x + 5y - 6z = This system can be written as a matrix equation of the form A X = B, where A is the coefficient matrix X is the column matrix of variables B is the column matrix of constant terms. In the given system, A = x, X = y z and B = 11-2

3 2. Write the matrix equation below. A X = B = 3. nter the three matrices on Derive using the Declare Matrix commands. Then use the Author command and the expression numbers to create the matrix equation. Use a period (.) to indicate matrix multiplication. (For example, if expression #10 is the coefficient matrix, expression #11 is the variable matrix, and expression #12 is the constant matrix, you would enter #10.#11=#12) Highlight the left hand side of the equation using the left arrow key and enter the Simplify command to show that this equation represents the original system. Write the result in the space below. 4. Before solving the matrix equation, find A -1. Use the Author command to enter the expression number of matrix A and typing ^-1. (For example, if matrix A is expression #10, enter #10^-1 ) Then press Simplify. A -1 = 5. Show that you have indeed found A -1 by multiplying A A -1 and A -1 A. + NOT: In A (-1) A, parentheses are required around the exponent. Otherwise DRIV thinks the period is a decimal point rather than matrix multiplication. A A -1 = A -1 A = 11-3

4 6. Recall that to solve the matrix equation A X = B, you must multiply by the inverse of A, A -1, on each side of the equation. A X = B A -1 A X = A -1 B I X = A -1 B where I is the identity matrix X = A -1 B + Recall that matrix multiplication is associative but not commutative, so you must multiply by A -1 on the left on both sides of the equation. 7. Now solve the matrix equation A X=B by multiplying each side of the matrix equation by A -1 using the Author command. Note that you do not actually have to compute A -1 to solve the matrix equation. For example, if expression #10 is the coefficient matrix, expression #11 is the variable matrix, and expression #12 is the constant matrix, you would enter #10^(-1).#10.#11=#10^(-1).#12 Highlight the left hand side of the resulting equation (use the left arrow key) and Simplify. Then highlight the right hand side of that resulting equation (right arrow key twice) and Simplify. Write the final result below. 8. Use the results of #7 to give the ordered triple solution to the original system of equations. 9. For each of the following systems, write down the system in matrix equation form. Then use the inverse of the coefficient matrix to solve the matrix equation, and write the solution. If the coefficient matrix has no inverse, say so, and then solve the system using the augmented matrix method. a. 3w - 4x + 5y + 6z = -21-7w + 2x - y + 8z = 16 w - x + y - 5z = 9 5w + 3x + 9y + z = -10 b. x + 2y + 3z = 4 4x + 5y + 6z = 7 7x + 8y + 9z =

5 Part 2 - Matrix Solutions on the TI-82 In this section, you will solve systems of equations using matrix equations and inverse matrices with the TI You can perform elementary row operations on the TI-82. The row operations are in the MATRX MATH window and include rowswap(, row+(, *row( and *row+(. These commands will not be used in this lab, but if you are interested in learning about them, you can find instructions in your TI-82 Guidebook. Consider the system of equations: x + 2y - 3z = 5 2x - 3y + 4z = 10 3x + 5y - 6z = This system can be written as a matrix equation of the form AX = B, where A is the coefficient matrix X is the column matrix of variables B is the column matrix of constant terms. In the given system, A = x, X = y z and B = 2. Write the matrix equation below. A X = B = 3. nter matrices A and B in the TI-82 as follows: To edit matrix A, > Press MATRX > Press for DIT > Press 1 for 1:[A] to edit matrix A Since A is a 3 x 3 matrix, > Type 3 and press NTR > Type 3 again and press NTR Now you can enter the numbers in the matrix by row, pressing NTR or using the arrow keys to move to the next entry. > Press 1 NTR 2 NTR etc. When you have made all your entries, > Press 2nd QUIT to return to the Home window 11-5

6 To see matrix A in the Home window, > Press MATRX 1 to enter the matrix name [A] in the Home window. > Press NTR to see the matrix A displayed in the Home window. > Use the above steps to help you enter matrix B. + NOT: You cannot enter matrix X since the TI-82 will not allow variable entries in a matrix. 4. Before solving the matrix equation, find A -1 with entries in fractional form. In the Home window, press MATRX 1 for matrix A. Then press the x -1 key. You will see [A]. Press MATH 1 for Frac and press NTR. + NOT: You must use the x -1 key. Typing ^-1 will not work. + NOT: The last column of entries is off the screen. To see the last column, press until you see the right-hand brackets of the matrix. A -1 = 5. Recall that to solve the matrix equation AX = B, you must multiply by the inverse of A, A -1, on each side of the equation. A X= B A -1 A X = A -1 B I X = A -1 B X = A -1 B where I is the identity matrix + Recall that matrix multiplication is associative but not commutative, so you must multiply by A -1 on the left on both sides of the equation. Thus the solution to the matrix equation AX = B is X = A -1 B. Note that you do not actually have to compute A -1 to solve the matrix equation. 6. Solve the matrix equation by computing A -1 B. Write your answers in fractional form. +HINT: Press MATRX 1 x -1 MATRX 2 MATH 1 x y z = 11-6

7 7. Try computing BA -1. What happens? Why? Is A -1 B= BA -1? Why or why not? 8. Use the results of #6 to give the ordered triple solution to the original system of equations. 9. For each of the following systems, write down the system in matrix equation form. Then use the inverse of the coefficient matrix to solve the matrix equation, and write the solution. If the coefficient matrix has no inverse, say so, and then solve the system using the augmented matrix method. a. 3w - 4x + 5y + 6z = -21-7w + 2x - y + 8z = 16 w - x + y - 5z = 9 5w + 3x + 9y + z = -10 b. x + 2y + 3z = 4 4x + 5y + 6z = 7 7x + 8y + 9z =

8 Part 3 - Modeling with Polynomial Functions You can use systems of equations to create polynomial models. 1. Find an equation of the parabola passing through the points (2, 5), (3, 9) and (5, -1) a. The general equation of a parabola is given by y = ax 2 + bx + c. Substituting the given points into this equation yields a system of three equations with unknowns a, b, and c. For example, substituting the point (2, 5) into y = ax 2 + bx + c yields the equation 5 = 4a + 2b + c preferrably written 4a + 2b + c = 5 Write the equations that result from substituting (3, 9) and (5, -1) into ax 2 + bx + c = y. b. Solve the system of three equations resulting from part a to find the coefficients a, b and c using an augmented matrix or matrix equations. Show the matrices used, and your results, below. c. Write the equation of the desired parabola using the coefficients found in #2. d. Graph the three points and the parabola to show that the points do fall on the parabola. Turn in a labeled copy of the graph with this lab. Recall that to create a linear model, you need to use two points. Similarly, to create a quadratic model, you need to use 3 points, since there are 3 coefficients to find. If you have more than 3 data points, you must choose 3 representative points from the data set. To create a cubic model, you need at 4 points; if you have more, then you must choose 4 representative points from the data set. Your model will fit exactly to the points you use to create the model, but may only come close to the rest of the data. 2. Use the same technique as in part 1 above to find a cubic equation, y = ax 3 + bx 2 + cx + d, that contains the points (0, 4), (1, 10), (2, 8), and (4, 4). Write your function in the space below. 11-8

9 Application: Just for the Halibut Pacific halibut are popular for commercial and sport fishing. These fish can vary in weight from several pounds to several hundred pounds. Anglers can estimate the weight of halibut they have caught based the length of the halibut. The following table gives the weight of a halibut based on its length. (Source: International Pacific Halibut Commission) Length of halibut (inches) Weight of halibut (pounds) Find a cubic equation that represents weight of a halibut as a function of its length. Give the points you used, your system of equations and your final model in the space below. 2. Graph your function along with the data. Turn in a labeled copy of the graph. 3. Use your model to answer the following: a. If you caught a halibut that was 2 feet long, how much would it weigh? b. Would a halibut that was twice as long as the one in part a (that is, 4 feet), also weigh twice as much? Justify your answer. c. Homer, Alaska holds a halibut fishing derby each year and is famous for its large halibut. The winner of the 1995 derby caught a halibut weighing a hefty pounds. stimate the length of the winning halibut to the nearest inch. 11-9

10 - Application: Acreage of Property: You wish to purchase a piece of property as shown. It is bordered by Descartes Drive to the east, Abel Avenue to the west, Hypatia Highway to the south and the Gaussian River to the north. You have only the map given below. The information provided to you by the seller says that the land is 1.25 acres, based on the measurements given. Use the method of curve fitting demonstrated in this lab and the method of finding area using trapezoids demonstrated in lab 10 to determine the area of the land in acres. Use at least 50 trapezoids. On your own paper, write a letter to the property owner stating whether you agree or disagree with the seller s estimate of the area of the land. Your letter should show step by step how you arrived at your results mathematically, with an explanation of each step. xplain why your answer is a better estimate of the acreage of the land than the seller s estimate. Include graphs or figures to assist your explanation. A B L A V N U 76 yds GAUSSIAN RIVR D S C A R T S 100 yds 50 yds 32 yds D R I V 20 yds 40 yds HYPATIA HIGHWAY 100 yards 11-10