10.3 Systems of Linear Equations: Determinants


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1 758 CHAPTER 10 Systems of Equtions nd Inequlities 10.3 Systems of Liner Equtions: Determinnts OBJECTIVES 1 Evlute 2 y 2 Determinnts 2 Use Crmer s Rule to Solve System of Two Equtions Contining Two Vriles 3 Evlute 3 y 3 Determinnts 4 Use Crmer s Rule to Solve System of Three Equtions Contining Three Vriles 5 Know Properties of Determinnts In the preeding setion, we desried method of using mtries to solve system of liner equtions. This setion dels with yet nother method for solving systems of liner equtions; however, it n e used only when the numer of equtions equls the numer of vriles. Although the method will work for ny system (provided tht the numer of equtions equls the numer of vriles), it is most often used for systems of two equtions ontining two vriles or three equtions ontining three vriles. This method, lled Crmer s Rule, is sed on the onept of determinnt. 1 Evlute 2 y 2 Determinnts If,,, nd d re four rel numers, the symol D = d is lled 2 y 2 determinnt. Its vlue is the numer d  ; tht is, D = d = d  The following devie my e helpful for rememering the vlue of 2 y 2 determinnt: d Minus d d EXAMPLE 1 Evluting 2 : 2 Determinnt Evlute:
2 SECTION 10.3 Systems of Liner Equtions: Determinnts 759 Algeri Solution = = = 15 Grphing Solution First, we enter the mtrix whose entries re those of the determinnt into the grphing utility nd nme it A. Using the determinnt ommnd, we otin the result shown in Figure 10. Figure 10 NOW WORK PROBLEM 7. 2 Use Crmer s Rule to Solve System of Two Equtions Contining Two Vriles Let s now see the role tht 2 y 2 determinnt plys in the solution of system of two equtions ontining two vriles. Consider the system x + y = s x + dy = t We shll use the method of elimintion to solve this system. Provided d Z 0 nd Z 0, this system is equivlent to the system dx + dy = sd x + dy = t Multiply y d. Multiply y. On sutrting the seond eqution from the first eqution, we get 1d  2x + 0 # y = sd  t x + dy = t Now the first eqution n e rewritten using determinnt nottion. d x = s t d If D = d = d  Z 0, we n solve for x to get s t d x = = d s t d D (3) Return now to the originl system. Provided tht Z 0 nd Z 0, the system is equivlent to x + y = s x + dy = t Multiply y. Multiply y.
3 760 CHAPTER 10 Systems of Equtions nd Inequlities On sutrting the first eqution from the seond eqution, we get x + y = s 0 # x + 1d  2y = t  s The seond eqution n now e rewritten using determinnt nottion. d y = s t If D = d = d  Z 0, we n solve for y to get y = s t d = s t D (4) Equtions (3) nd (4) led us to the following result, lled Crmer s Rule. Crmer s Rule for Two Equtions Contining Two Vriles The solution to the system of equtions x + y = s x + dy = t (5) is given y x = s t d, y = d s t d (6) provided tht D = d = d  Z 0 In the derivtion given for Crmer s Rule ove, we ssumed tht none of the numers,,, nd d ws 0. In Prolem 60 you will e sked to omplete the proof under the less stringent ondition tht D = d  Z 0. Now look refully t the pttern in Crmer s Rule. The denomintor in the solution (6) is the determinnt of the oeffiients of the vriles. x + y = s D = x + dy = t d
4 SECTION 10.3 Systems of Liner Equtions: Determinnts 761 In the solution for x, the numertor is the determinnt, denoted y D x, formed y repling the entries in the first olumn (the oeffiients of x) of D y the onstnts on the right side of the equl sign. s D x = t In the solution for y, the numertor is the determinnt, denoted y D y, formed y repling the entries in the seond olumn (the oeffiients in y) of D y the onstnts on the right side of the equl sign. D y = Crmer s Rule then sttes tht, if D Z 0, d s t x = D x D, y = D y D (7) EXAMPLE 2 Solving System of Liner Equtions Using Determinnts Use Crmer s Rule, if pplile, to solve the system 3x  2y = 4 6x + y = 13 Algeri Solution The determinnt D of the oeffiients of the vriles is 32 D = 6 1 = = 15 Beuse D Z 0, Crmer s Rule (7) n e used. 42 x = D x D = y = D y 15 D = = = = 30 = = 2 = 1 Grphing Solution We enter the oeffiient mtrix into our grphing utility. Cll it A nd evlute det3a4. Sine det3a4 Z 0, we n use Crmer s Rule. We enter the mtries D x nd D y into our grphing utility nd ll them B nd C, respetively. Finlly, we find x det3b4 det3c4 y lulting nd y y lulting det3a4 det3a4. The results re shown in Figure 11. Figure 11 The solution is x = 2, y = 1. In ttempting to use Crmer s Rule, if the determinnt D of the oeffiients of the vriles is found to equl 0 (so tht Crmer s Rule is not pplile), then the system either is inonsistent or hs infinitely mny solutions. NOW WORK PROBLEM 15.
5 762 CHAPTER 10 Systems of Equtions nd Inequlities 3 Evlute 3 y 3 Determinnts To use Crmer s Rule to solve system of three equtions ontining three vriles, we need to define 3 y 3 determinnt. A 3 y 3 determinnt is symolized y (8) in whih 11, 12, Á, re rel numers. As with mtries, we use doule susript to identify n entry y inditing its row nd olumn numers. For exmple, the entry 23 is in row 2, olumn 3. The vlue of 3 y 3 determinnt my e defined in terms of 2 y 2 determinnts y the following formul: Minus = q q q 2 y 2 2 y 2 2 y 2 determinnt determinnt determinnt left fter left fter left fter removing row removing row removing row nd olumn nd olumn nd olumn ontining ontining ontining (9) The 2 y 2 determinnts shown in formul (9) re lled minors of the 3 y 3 determinnt. For n n y n determinnt, the minor M ij of entry ij is the determinnt resulting from removing the ith row nd jth olumn. EXAMPLE 3 Finding Minors of 3 y 3 Determinnt For the determinnt A = , find: () M 12 () M 23 Solution () M 12 is the determinnt tht results from removing the first row nd seond olumn from A A = M 12 = = = () M 23 is the determinnt tht results from removing the seond row nd third olumn from A. A = M 23 = = = 12
6 SECTION 10.3 Systems of Liner Equtions: Determinnts 763 Referring k to formul (9), we see tht eh element is multiplied y its minor, ut sometimes this term is dded nd other times, sutrted. To determine whether to dd or sutrt term, we must onsider the oftor. For n n y n determinnt A, the oftor of entry ij, denoted y A ij, is given y ij where M ij is the minor of entry ij. A ij = 112 i + j M ij The exponent of 112 i + j is the sum of the row nd olumn of the entry ij, so if i + j is even, 112 i + j will equl 1, nd if i + j is odd, 112 i + j will equl 1. To find the vlue of determinnt, multiply eh entry in ny row or olumn y its oftor nd sum the results.this proess is referred to s expnding ross row or olumn. For exmple, the vlue of the 3 y 3 determinnt in formul (9) ws found y expnding ross row 1. If we hoose to expnd down olumn 2, we otin = æ Expnd down olumn 2. If we hoose to expnd ross row 3, we otin = æ Expnd ross row 3. It n e shown tht the vlue of determinnt does not depend on the hoie of the row or olumn used in the expnsion. However, expnding ross row or olumn tht hs n element equl to 0 redues the mount of work needed to ompute the vlue of the determinnt. EXAMPLE 4 Evluting 3 : 3 Determinnt Find the vlue of the 3 y 3 determinnt: Solution We hoose to expnd ross row = = = = = 138
7 764 CHAPTER 10 Systems of Equtions nd Inequlities We ould lso find the vlue of the 3 y 3 determinnt in Exmple 4 y expnding down olumn = = = = 138 Evluting 3 * 3 determinnts on grphing utility follows the sme proedure s evluting 2 * 2 determinnts. NOW WORK PROBLEM Use Crmer s Rule to Solve System of Three Equtions Contining Three Vriles Consider the following system of three equtions ontining three vriles. 11 x + 12 y + 13 z = 1 21 x + 22 y + 23 z = 2 31 x + 32 y + 33 z = 3 It n e shown tht if the determinnt D of the oeffiients of the vriles is not 0, tht is, if D = Z then the unique solution of system (10) is given y Crmer s Rule for Three Equtions Contining Three Vriles (10) x = D x y = D y z = D z D D D where D x = D y = D z = Do you see the similrity of this pttern nd the pttern oserved erlier for system of two equtions ontining two vriles? EXAMPLE 5 Using Crmer s Rule Use Crmer s Rule, if pplile, to solve the following system: 2x + y  z = 3 x + 2y + 4z = 3 x  2y  3z = 4 (3)
8 Solution SECTION 10.3 Systems of Liner Equtions: Determinnts 765 The vlue of the determinnt D of the oeffiients of the vriles is D = = = Beuse D Z 0, we proeed to find the vlues of D x, D y, nd D z D y = = = = = D z = = As result, = = D x = = = = 15 = = 5 x = D x D = 15 5 = 3, y = D y D = The solution is x = 3, y = 2, z = If the determinnt of the oeffiients of the vriles of system of three liner equtions ontining three vriles is 0, then Crmer s Rule is not pplile. In suh se, the system either is inonsistent or hs infinitely mny solutions. Solving systems of three equtions ontining three vriles using Crmer s Rule on grphing utility follows the sme proedure s tht for solving systems of two equtions ontining two vriles. NOW WORK PROBLEM 33. = 2, z = D z D = 5 5 = 1 5 Know Properties of Determinnts Determinnts hve severl properties tht re sometimes helpful for otining their vlue. We list some of them here. The vlue of determinnt hnges sign if ny two rows (or ny two olumns) re interhnged. (11) Proof for 2 y 2 Determinnts d = d  nd d =  d = 1d  2
9 766 CHAPTER 10 Systems of Equtions nd Inequlities EXAMPLE 6 Demonstrting (11) = 64 = = 46 = 2 If ll the entries in ny row (or ny olumn) equl 0, the vlue of the determinnt is 0. (12) Proof Expnd ross the row (or down the olumn) ontining the 0 s. If ny two rows (or ny two olumns) of determinnt hve orresponding entries tht re equl, the vlue of the determinnt is 0. (13) You re sked to prove this result for 3 y 3 determinnt in whih the entries in olumn 1 equl the entries in olumn 3 in Prolem 63. EXAMPLE 7 Demonstrting (13) = = = = 0 If ny row (or ny olumn) of determinnt is multiplied y nonzero numer k, the vlue of the determinnt is lso hnged y ftor of k. (14) You re sked to prove this result for 3 y 3 determinnt using row 2 in Prolem 62. EXAMPLE 8 Demonstrting (14) = 68 = 2 k 2k 4 6 = 6k  8k = 2k = k122 = k If the entries of ny row (or ny olumn) of determinnt re multiplied y nonzero numer k nd the result is dded to the orresponding entries of nother row (or olumn), the vlue of the determinnt remins unhnged. (15) In Prolem 64, you re sked to prove this result for 3 y 3 determinnt using rows 1 nd 2.
10 SECTION 10.3 Systems of Liner Equtions: Determinnts 767 EXAMPLE 9 Demonstrting (15) = : = 14 æ Multiply row 2 y 2 nd dd to row 1.
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