Matrices and Systems of Linear Equations

Transcription

1 Chapter The variable x has now been eliminated from the first and third equations. Next, we eliminate x3 from the first and second equations and leave x3, with coefficient, in the third equation: System: Augmented Matrix: (-/3)E3: (- /3)R3: u u - 5x3-3 X xz + x3 X3 R + 5R3: E + SE3: 7 XJ Xz + x3 X3 R - R3: E - E3: XJ 7 Xz - X The last system above clearly has a unique solution given by x 7, x -, and x3. Because the final system is equivalent to the original given system, both systems have the same solution. llllm The reduction process used in the preceding example is known as Gauss-Jordan elimination and will be explained in Section.. Note the advantage of the shorthand notation provided by matrices. B ecause we do not need to list the variables, the sequence of steps in the right-hand column is easier to perform and record. Example 7 illustrates that row equivalent augmented matrices represent equivalent systems of equations. The following corollary to Theorem states this in mathematical terms. COl\O LLA l Y Iii! EXERCISES Which of the equations in Exercises -6 are linear?. XX + Xz. XJ + X x - x sin x + cos x - n) Suppose [A I b and [CI d are augmented matrices, each representing a different (m x system of linear equations. If [A I b and [CI d are row equivalent matrices, then the two systems are also equivalent x - x sin x + cos x 6. nx + v'7x../3 5. Jx l lxl In Exercises 7-, coefficients are given for a system of the form (). Display the system and verify that the given values constitute a solution. 7.al, b 7, a 3, b ; a 4, X, a -, Xz

2 . Introduction to 3 8. a 6, az, XJ, a -, a3 4, X -l, a3, b 4, X3 9. a l, al, a 3, a3 -, a3, b, b3-3; XJ, X -. a, a 3, b 9, b 8; a I, bi 4; az 4, b -, az 4, an, XJ, Xz 3 In Exercises L-4, sketch a graph for each equation to determine whether the system has a unique solution, no solution, or infinitely many solutions.. x + y 5. x - y - x- yl x- y 3. 3x + y 6 4. x + y 5-6x- 4y- x- yl x + 3y 9 5. The ( x 3) system of linear equations ax + by + cz d ax + by + cz dz is represented geometrically by two planes. How are the planes related when: a) The system has no solution? b) The system has infinitely many solutions? Is it possible for the system to have a unique solution? Explain. In Exercises 6--8, determine whether the given ( x 3) system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line; that is, give equations of the form x at + b, y ct + d, z et+ f. 6. x + Xz + X X + x - X3 -x + Xz - X3 X + Xz + X x + 3x - x3 - x +6x-4x3-9. Display the ( x3) matrix A (a;j), wherea, a, a3 6, az 4, an 3, and a3 8.. Display the ( x4) matrix C (c;j), where c3 4, c, c, c 4 I, C, c4 3, Ct!, and C3 7.. Display the (3x3) matrix Q (qu), whereq3 l, q3, q, q3-3, q, q33, qz, q 4, and q3 3.. Suppose the matrix C in Exercise is the augmented matrix for a system of linear equations. Display the system. 3. Repeat Exercise for the matrices in Exercises 9 and. In Exercises 4-9, display the coefficient matrix A and the augmented matrix B for the given system. 4. Xt - X - 5. Xt + Xz - X3 Xt + X 3 x - X3 l 6. x + 3x - X3 7. x + x + x3 6 x + 5x + X3 5 3x + 4x - X3 5 X + X + X3 3 -xi + Xz + X3 8. x + x - 3x3 - x + x - 5x3 - -x- 3x+7x XJ + Xz + X3 x + 3x + X3 x - xz + 3x3 In Exercises 3-36, display the augmented matrix for the given system. Use elementary operations on equations to obtain an equivalent system of equations in which x appears in the first equation with coefficient one and has been eliminated from the remaining equations. Simultaneously, perform the corresponding elementary row operations on the augmented matrix. 3. x + 3x 6 3. x + x - x3 4x - X 7 X + X + x3 -X + X 3. Xz + X3 4 X - X + x3 x + Xz - x x, +x +x3 - x4 -X + Xz - X3 + X4 3 -X + Xz + X3 - X4 35. Xt + xz - X3 + X4 -xi + xz + 7x3 - X4 36. x +x X -Xz 3x +x Xz + X3 - X XJ + Xz 9 XJ -Xz 7 3x +x Consider the equation x - 3x + X3 - X a) In the six different possible combinations, set any two of the variables equal to and graph the equation in terms of the other two. b) What type of graph do you always get when you set two of the variables equal to two fixed constants? c) What is one possible reason the equation in formula () is called linear?

3 6 Chapter A D D I N G I NT-EG-E Rf Mathematical folklore has it that Gauss discovered the formula n n(n + )/ when he was only ten years old. To occupy time, his teacher asked the students to add the integers from to. Gauss immediately wrote an answer and turned his slate over. To his teacher's amazement, Gauss had the only correct answer in the class. Young Gauss had recognized that the numbers could be put in 5 sets of pairs such that the sum of each pair was : (5 + 5 ) + (49 + 5) + ( ) ( + ) 5() 55. Soon his brilliance was brought to the attention of the Duke of Brunswick, who thereafter sponsored the education of Gauss. EXERCISES Consider the matrices in Exercises -.. a) Either state that the matrix is in echelon form or use elementary row operations to transform it to echelon form. b) If the matrix is in echelon form, transform it to reduced echelon form In Exercises -35, solve the system by transforming the augmented matrix to reduced echelon form. In Exercises -, each ofthe given matrices represents the augmented matrix for a system of linear equations. In each exercise, display the solution set or state that the system is inconsistent.. 3. x - 3x 5-4x + 6x - X - x 3 x - 4x

4 . Echelon Form and Gauss-Jordan Elimination 7 4. Xi - Xz + X3 3 xi +x -4x Xi + X 3xi +3x 6 6. Xi - Xz + X3 4 xi - x + 3x3 7. Xi + Xz - X3-3xi - 3x + 3x xi + 3x - 4x3 3 xi - x - x3 - -xi + l6x + x xi + x - x3 I xi -xz + 7x3 8 -xi +x - 5x Xi + X - X5 I x + x3 + X4 + 3xs - X3 + X4 + X5 + x3 + X4 - xs xi + X + 3x3 - X4 + X5 3xi - x + 4x3 + X4 + xs 3. Xi + Xz 33. Xi + Xz Xi -Xz 3 xi +x xi +x I xi +4x -xi -x - Xi - X 3 x + x 35. Xi - X - X3 X + X3 x + x3 3 In Exercises 36--4, find all values a for which the system has no solution. 36. Xi +Xz -3 axi - x xi +4x a 3xi + 6x 5 4. xi + ax 6 ax +ax X +3x 4 x +6x a 39. 3xi +ax 3 axi + 3x 5 In Exercises 4 and 4, find all values a and fj where S a S n and S fj S n. 4. cosa + 4sinfJ 3 3 cos a - 5 sin fj - I 4. cos a - sin fj cos a + 8 sin f Describe the solution set of the following system in terms of x3 : X +x 5. For x, x, X3 in the solution set: a) Find the maximum value of x3 such that x '.:: and x ::. b) Find the maximum value of y xi - 4x + x3 subject to x :: and Xz '.::. c) Find the minimum value of y (x - ) + (x + 3) + (x3 + ) with no restriction on x or x. [Hint: Regard y as a function of x3 and set the derivative equal to O; then apply the second-derivative test to verify that you have found a minimum. 44. Let A and I be as follows: A[ : l / [ l Prove that if b - cd :f, then A is row equivalent to /. 45. As in Fig..4, display all the possible configurations for a ( x 3) matrix that is in echelon form. [Hint: There are seven such configurations. Consider the various positions that can be occupied by one, two, or none of the symbols. 46. Repeat Exercise 45 for a (3 x ) matrix, for a (3 x 3) matrix, and for a (3 x 4) matrix. 47. Consider the matrices B and C: B C [ [, 3 By Exercise 44, B and C are both row equivalent to matrix I in Exercise 44. Determine elementary row operations that demonstrate that B is row equivalent to C. 48. Repeat Exercise 47 for the matrices B, C [ 4 [ 49. A certain three-digit number N equals fifteen times the sum of its digits. If its digits are reversed, the re sulting number exceeds N by 396. The one's digit is one larger than the sum of the other two. Give a linear system of three equations whose three unknowns are the digits of N. Solve the system and find N. 5. Find the equation of the parabola, y ax +bx +c, that passes through the points (-, 6), (, 4), and (, 9). [Hint: For each point, give a linear equation in a, b, and c. 5. Three people play a game in which there are always two winners and one loser. They have the.

5 8 Chapter understanding that the loser gives each winner an amount equal to what the winner already has. After three games, each has lost just once and each has $4. With how much money did each begin? 5. Find three nu mbers whose sum is 34 when the sum of the first and second is 7, and the sum of the second and third is. 53. A zoo charges $6 for adults, $3 for students, and $.5 for children. One morning 79 people enter and pay a total of $7. Determine the possible numbers of adults, students, and children. 54. Find a cubic polynomial, p (x) a+bx +cx +dx 3, such that p(i) 5, p' ( I) 5, p () 7, and p ' (). In Exercises 55-58, use Eq. () to find the formula for the sum. If available, use l inear algebra software for Exercises 57 and I++3+ +n CONSISTENT SYSTEMS OF LINEAR EQUATIONS We saw in Section. that a system of linear equations may have a unique solution, infinitely many solutions, or no solution. In thi s section and in later sections, it will be shown that with certain added bits of information we can, without solving the system, either eliminate one of the three possible outcomes or determine precisely what the outcome will be. This will be important l ater when situations will arise in which we are not interested in obtaining a specific solution, but we need to know only how many solutions there are. To illustrate, consider the general ( x l i near system 3) Geometrically, the system is represented by two planes, and a solution corresponds to a point in the intersection of the planes. The two planes may be parallel, they may be coincident (the same plane), or they may intersect in a line. Thus the system i s either inconsi stent o r has infinitely many solutions; the existence o f a unique solution is impossible. Solution Possibilities for a Consistent Linear System We begin our analysis by considering the (m x n) system of l inear equations a, X + ax + + a,,,x,, ax + ax llx b b a X + GX + + GmX bm. Our goal is to deduce as m uch i nformation as possible about the solution set of system ( ) without actuall y solving the system. To that end, let [A I b denote the augmented matrix for system ( I). We know we can use row operations to transform the [ m x ( + I) matrix [A I b to a row equivalent matrix [ c I dj where r c I d is in reduced echelon form. Hence, instead of trying to