Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
758 CHAPTER 10 10.3 Systems of Equations and Inequalities Systems of Linear Equations: Determinants OBJECTIVES 1 2 3 4 5 Evaluate 2 by 2 Determinants Use Cramer’s Rule to Solve a System of Two Equations Containing Two Variables Evaluate 3 by 3 Determinants Use Cramer’s Rule to Solve a System of Three Equations Containing Three Variables Know Properties of Determinants In the preceding section, we described a method of using matrices to solve a system of linear equations. This section deals with yet another method for solving systems of linear equations; however, it can be used only when the number of equations equals the number of variables. Although the method will work for any system (provided that the number of equations equals the number of variables), it is most often used for systems of two equations containing two variables or three equations containing three variables. This method, called Cramer’s Rule, is based on the concept of a determinant. 1 Evaluate 2 by 2 Determinants ✓ If a, b, c, and d are four real numbers, the symbol D = ` a b ` c d is called a 2 by 2 determinant. Its value is the number ad - bc; that is, D = ` a b ` = ad - bc c d (1) The following device may be helpful for remembering the value of a 2 by 2 determinant: bc a b c d ad bc Minus EXAMPLE 1 ad Evaluating a 2 : 2 Determinant Evaluate: ` 3 6 -2 ` 1 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall SECTION 10.3 Algebraic Solution 3 ` 6 759 Graphing Solution First, we enter the matrix whose entries are those of the determinant into the graphing utility and name it A. Using the determinant command, we obtain the result shown in Figure 10. -2 ` = 132112 - 1621-22 1 = 3 - 1-122 = 15 Systems of Linear Equations: Determinants Figure 10 NOW WORK PROBLEM 7. 2 Use Cramer’s Rule to Solve a System of Two Equations ✓ Containing Two Variables Let’s now see the role that a 2 by 2 determinant plays in the solution of a system of two equations containing two variables. Consider the system ax + by = s cx + dy = t b (1) (2) (2) We shall use the method of elimination to solve this system. Provided d Z 0 and b Z 0, this system is equivalent to the system b adx + bdy = sd bcx + bdy = tb (1) Multiply by d. (2) Multiply by b. On subtracting the second equation from the first equation, we get b 1ad - bc2x + 0 # y = sd - tb bcx + bdy = tb (1) (2) Now the first equation can be rewritten using determinant notation. ` If D = ` a b s b `x = ` ` c d t d a b ` = ad - bc Z 0, we can solve for x to get c d x = ` s b ` t d a b ` ` c d ` = s b ` t d (3) D Return now to the original system (2). Provided that a Z 0 and c Z 0, the system is equivalent to b acx + bcy = cs acx + ady = at (1) Multiply by c. (2) Multiply by a. Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall 760 CHAPTER 10 Systems of Equations and Inequalities On subtracting the first equation from the second equation, we get b acx + bcy = cs 0 # x + 1ad - bc2y = at - cs (1) (2) The second equation can now be rewritten using determinant notation. ` If D = ` a b a s `y = ` ` c d c t a b ` = ad - bc Z 0, we can solve for y to get c d ` y = a s ` c t a b ` ` c d ` a s ` c t = (4) D Equations (3) and (4) lead us to the following result, called Cramer’s Rule. Theorem Cramer’s Rule for Two Equations Containing Two Variables The solution to the system of equations b ax + by = s cx + dy = t (1) (2) (5) is given by x = ` s b ` t d a b ` ` c d , y = ` a s ` c t a b ` ` c d (6) provided that D = ` a b ` = ad - bc Z 0 c d In the derivation given for Cramer’s Rule above, we assumed that none of the numbers a, b, c, and d was 0. In Problem 60 you will be asked to complete the proof under the less stringent condition that D = ad - bc Z 0. Now look carefully at the pattern in Cramer’s Rule. The denominator in the solution (6) is the determinant of the coefficients of the variables. b ax + by = s cx + dy = t D = ` a b ` c d Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall SECTION 10.3 Systems of Linear Equations: Determinants 761 In the solution for x, the numerator is the determinant, denoted by Dx , formed by replacing the entries in the first column (the coefficients of x) of D by the constants on the right side of the equal sign. Dx = ` s b ` t d In the solution for y, the numerator is the determinant, denoted by Dy , formed by replacing the entries in the second column (the coefficients in y) of D by the constants on the right side of the equal sign. Dy = ` a s ` c t Cramer’s Rule then states that, if D Z 0, x = EXAMPLE 2 Dx , D y = Dy (7) D Solving a System of Linear Equations Using Determinants Use Cramer’s Rule, if applicable, to solve the system b 3x - 2y = 4 6x + y = 13 (1) (2) Algebraic Solution Graphing Solution The determinant D of the coefficients of the variables is We enter the coefficient matrix into our graphing utility. Call it A and evaluate det3A4. Since det3A4 Z 0, we can use Cramer’s Rule. We enter the matrices Dx and Dy into our graphing utility and call them B and C, respectively. Finally, we find x det3B4 det3C4 . by calculating and y by calculating det3A4 det3A4 The results are shown in Figure 11. D = ` 3 6 -2 ` = 132112 - 1621-22 = 15 1 Because D Z 0, Cramer’s Rule (7) can be used. ` 4 13 -2 ` 1 Dx = D 15 142112 - 11321-22 = 15 30 = 15 = 2 x = y = Dy ` 3 6 4 ` 13 = D 15 1321132 - 162142 = 15 15 = 15 = 1 Figure 11 The solution is x = 2, y = 1. In attempting to use Cramer’s Rule, if the determinant D of the coefficients of the variables is found to equal 0 (so that Cramer’s Rule is not applicable), then the system either is inconsistent or has infinitely many solutions. NOW WORK PROBLEM 15. Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall 762 CHAPTER 10 Systems of Equations and Inequalities 3 Evaluate 3 by 3 Determinants ✓ To use Cramer’s Rule to solve a system of three equations containing three variables, we need to define a 3 by 3 determinant. A 3 by 3 determinant is symbolized by a12 a22 a32 a11 3 a21 a31 a13 a23 3 a33 (8) in which a11 , a12 , Á , are real numbers. As with matrices, we use a double subscript to identify an entry by indicating its row and column numbers. For example, the entry a23 is in row 2, column 3. The value of a 3 by 3 determinant may be defined in terms of 2 by 2 determinants by the following formula: Minus a11 3 a21 a31 a12 a22 a32 a13 a a23 3 = a11 ` 22 a32 a33 a23 ∂ a ` - a12 ` 21 a33 a31 q 2 by 2 determinant left after removing row and column containing a11 a 23 a ` + a13 ` 21 a 33 a31 q 2 by 2 determinant left after removing row and column containing a12 a22 ` a32 (9) q 2 by 2 determinant left after removing row and column containing a13 The 2 by 2 determinants shown in formula (9) are called minors of the 3 by 3 determinant. For an n by n determinant, the minor Mij of entry aij is the determinant resulting from removing the ith row and jth column. EXAMPLE 3 Finding Minors of a 3 by 3 Determinant 2 3 For the determinant A = -2 0 Solution -1 5 6 3 1 3 , find: -9 (a) M12 (b) M23 (a) M12 is the determinant that results from removing the first row and second column from A. 2 A = 3 -2 0 -1 5 6 3 13 -9 M12 = ` -2 0 1 ` = 1-221-92 - 102112 = 18 -9 (b) M23 is the determinant that results from removing the second row and third column from A. 2 A = 3 -2 0 -1 5 6 3 13 -9 M23 = ` 2 0 -1 ` = 122162 - 1021-12 = 12 6 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall SECTION 10.3 763 Systems of Linear Equations: Determinants Referring back to formula (9), we see that each element aij is multiplied by its minor, but sometimes this term is added and other times, subtracted. To determine whether to add or subtract a term, we must consider the cofactor. For an n by n determinant A, the cofactor of entry aij , denoted by A ij , is given by A ij = 1-12i + jMij where Mij is the minor of entry aij . The exponent of 1-12i + j is the sum of the row and column of the entry aij , so if i + j is even, 1-12i + j will equal 1, and if i + j is odd, 1-12i + j will equal -1. To find the value of a determinant, multiply each entry in any row or column by its cofactor and sum the results. This process is referred to as expanding across a row or column. For example, the value of the 3 by 3 determinant in formula (9) was found by expanding across row 1. If we choose to expand down column 2, we obtain a11 3 a21 a31 a12 a22 a32 a13 a a23 3 = 1-121 + 2 a12 ` 21 a31 a33 a23 a ` + 1-122 + 2a22 ` 11 a33 a31 a13 a ` + 1-123 + 2a32 ` 11 a33 a21 a13 ` a23 a13 a ` + 1-123 + 3a33 ` 11 a23 a21 a12 ` a22 æ Expand down column 2. If we choose to expand across row 3, we obtain a11 3 a21 a31 a12 a22 a32 a13 a a23 3 = 1-123 + 1 a31 ` 12 a22 a33 a13 a ` + 1-123 + 2a32 ` 11 a23 a21 æ Expand across row 3. It can be shown that the value of a determinant does not depend on the choice of the row or column used in the expansion. However, expanding across a row or column that has an element equal to 0 reduces the amount of work needed to compute the value of the determinant. EXAMPLE 4 Evaluating a 3 : 3 Determinant Find the value of the 3 by 3 determinant: 3 34 8 Solution 4 6 -2 -1 23 3 We choose to expand across row 1. 3 34 8 4 6 -2 -1 6 2 3 = 1-121 + 1 3 ` -2 3 2 4 ` + 1-121 + 24 ` 3 8 2 4 ` + 1-121 + 31-12 ` 3 8 = 3118 + 42 - 4112 - 162 + 1-121-8 - 482 = 31222 - 41-42 + 1-121-562 = 66 + 16 + 56 = 138 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall 6 ` -2 764 CHAPTER 10 Systems of Equations and Inequalities We could also find the value of the 3 by 3 determinant in Example 4 by expanding down column 3. 3 34 8 4 6 -2 -1 4 2 3 = 1-121 + 31-12 ` 8 3 6 3 ` + 1-122 + 32 ` -2 8 4 3 ` + 1-123 + 33 ` -2 4 4 ` 6 = -11-8 - 482 - 21-6 - 322 + 3118 - 162 = 56 + 76 + 6 = 138 Evaluating 3 * 3 determinants on a graphing utility follows the same procedure as evaluating 2 * 2 determinants. NOW WORK PROBLEM 11. 4 Use Cramer’s Rule to Solve a System of Three Equations ✓ Containing Three Variables Consider the following system of three equations containing three variables. a11x + a12 y + a13 z = c1 c a21x + a22 y + a23 z = c2 a31x + a32 y + a33 z = c3 (10) It can be shown that if the determinant D of the coefficients of the variables is not 0, that is, if a11 a12 a13 D = 3 a21 a22 a23 3 Z 0 a31 a32 a33 then the unique solution of system (10) is given by Cramer’s Rule for Three Equations Containing Three Variables x = Dx D y = Dy D z = Dz D where c1 Dx = 3 c2 c3 a12 a22 a32 a13 a23 3 a33 a11 Dy = 3 a21 a31 c1 c2 c3 a13 a23 3 a33 a11 Dz = 3 a21 a31 a12 a22 a32 c1 c2 3 c3 Do you see the similarity of this pattern and the pattern observed earlier for a system of two equations containing two variables? EXAMPLE 5 Using Cramer’s Rule Use Cramer’s Rule, if applicable, to solve the following system: 2x + y - z = 3 c -x + 2y + 4z = -3 x - 2y - 3z = 4 (1) (2) (3) Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall Solution 2 D = 3 -1 1 765 Systems of Linear Equations: Determinants SECTION 10.3 The value of the determinant D of the coefficients of the variables is 1 2 -2 -1 2 4 -1 4 3 = 1-121 + 1 2 ` ` + 1-121 + 21 ` -2 -3 1 -3 = 2122 - 11-12 + 1-12102 = 4 + 1 = 5 4 -1 ` + 1-121 + 31-12 ` -3 1 2 ` -2 Because D Z 0, we proceed to find the values of Dx , Dy , and Dz . 3 Dx = 3 -3 4 1 2 -2 -1 2 4 -3 4 3 = 1-121 + 1 3 ` ` + 1-121 + 21 ` -2 -3 4 -3 = 3122 - 11-72 + 1-121-22 = 15 4 -3 ` + 1-121 + 31-12 ` -3 4 2 ` -2 2 Dy = 3 -1 1 3 -3 4 4 -1 ` + 1-121 + 31-12 ` -3 1 -3 ` 4 2 Dz = 3 -1 1 1 2 -2 -1 43 = -3 = = 3 -3 3 = 4 = 1-121 + 1 2 ` -3 4 4 -1 ` + 1-121 + 23 ` -3 1 21-72 - 31-12 + 1-121-12 -14 + 3 + 1 = -10 1-121 + 1 2 ` 2 -2 -3 -1 ` + 1-121 + 21 ` 4 1 -3 -1 ` + 1-121 + 33 ` 4 1 2 ` -2 2122 - 11-12 + 3102 = 5 As a result, x = Dx 15 = = 3, D 5 y = Dy D = -10 = -2, 5 z = Dz D = 5 = 1 5 The solution is x = 3, y = -2, z = 1. If the determinant of the coefficients of the variables of a system of three linear equations containing three variables is 0, then Cramer’s Rule is not applicable. In such a case, the system either is inconsistent or has infinitely many solutions. Solving systems of three equations containing three variables using Cramer’s Rule on a graphing utility follows the same procedure as that for solving systems of two equations containing two variables. NOW WORK PROBLEM 33. 5 Know Properties of Determinants ✓ Determinants have several properties that are sometimes helpful for obtaining their value. We list some of them here. Theorem The value of a determinant changes sign if any two rows (or any two columns) are interchanged. (11) Proof for 2 by 2 Determinants ` a b ` = ad - bc and c d ` c d ` = bc - ad = -1ad - bc2 a b Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall ■ 766 CHAPTER 10 Systems of Equations and Inequalities EXAMPLE 6 Demonstrating Theorem (11) ` Theorem 3 1 4 ` =6-4=2 2 ` 1 3 2 ` = 4 - 6 = -2 4 If all the entries in any row (or any column) equal 0, the value of the determinant is 0. (12) ■ Proof Expand across the row (or down the column) containing the 0’s. Theorem If any two rows (or any two columns) of a determinant have corresponding entries that are equal, the value of the determinant is 0. (13) You are asked to prove this result for a 3 by 3 determinant in which the entries in column 1 equal the entries in column 3 in Problem 63. EXAMPLE 7 Demonstrating Theorem (13) 1 31 4 Theorem 2 2 5 3 2 3 1 3 1 3 3 = 1-121 + 11 ` ` + 1-121 + 22 ` ` + 1-121 + 33 ` 5 6 4 6 4 6 = 11-32 - 21-62 + 31-32 = -3 + 12 - 9 = 0 2 ` 5 If any row (or any column) of a determinant is multiplied by a nonzero number k, the value of the determinant is also changed by a factor of k. (14) You are asked to prove this result for a 3 by 3 determinant using row 2 in Problem 62. EXAMPLE 8 Demonstrating Theorem (14) ` ` Theorem 1 4 2 ` = 6 - 8 = -2 6 k 2k 1 ` = 6k - 8k = -2k = k1-22 = k ` 4 6 4 2 ` 6 If the entries of any row (or any column) of a determinant are multiplied by a nonzero number k and the result is added to the corresponding entries of another row (or column), the value of the determinant remains unchanged. (15) In Problem 64, you are asked to prove this result for a 3 by 3 determinant using rows 1 and 2. Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall SECTION 10.3 EXAMPLE 9 Systems of Linear Equations: Determinants 767 Demonstrating Theorem (15) ` 3 5 4 ` = -14 2 ` 3 5 4 -7 `:` 2 5 0 ` = -14 2 æ Multiply row 2 by -2 and add to row 1. Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall