CHAPTER 10 Matrices and Determinants

Transcription

1 CHAPTER Matrices and Determinants Section. Matrices and Sstems of Equations Section. Operations with Matrices Section. The Inverse of a Square Matri Section. The Determinant of a Square Matri Section. Applications of Matrices and Determinants Review Eercises Problem Solving Practice Test

2 CHAPTER Matrices and Determinants Section. Matrices and Sstems of Equations You should be able to use elementar row operations to produce a row-echelon form (or reduced row-echelon form) of a matri.. Interchange two rows.. Multipl a row b a nonzero constant.. Add a multiple of one row to another row. You should be able to use either Gaussian elimination with back-substitution or Gauss- Jordan elimination to solve a sstem of linear equations. Vocabular Check. matri. square. main diagonal. row; column. augmented. coefficient. row-equivalent. reduced row-echelon form. Gauss-Jordan elimination. Since the matri has one row and two columns, its order is.. Since the matri has one row and four columns, its order is.. Since the matri has three rows and one column, its order is.. Since the matri has three rows and four columns, its order is.. Since the matri has two rows and two columns, its order is.. Since the matri has two rows and three columns, its order is.... z z.. z z z.. z z.. z z z z

3 Chapter Matrices and Determinants. z z. z z w z z. z z z z w w w w. R R. R. R R R R R. R R R R R.. Add times Row to Row. Add times Row to Row.. Interchange Row and Row. Then add times the new Row to Row.. Add times Row to Row. Add times Row to Row.. (a) (c) (d) (e) This matri is in reduced row-echelon form.

4 Section. Matrices and Sstems of Equations.. (a) (d) (e). This matri is in reduced row-echelon form. (c) (f) This matri is in reduced row-echelon form. This matri is in reduced row-echelon form... The first nonzero entries in Rows and are not. The matri is not in row-echelon form. This matri is in row-echelon form.. R R R R. R R R R R R R R. R R R R R R. R R R R R R. Use the reduced row-echelon form feature of a graphing utilit.. Use the reduced row-echelon form feature of a graphing utilit.

5 Chapter Matrices and Determinants. Use the reduced row-echelon form feature of a graphing utilit.. Use the reduced row-echelon form feature of a graphing utilit.. Use the reduced row-echelon form feature of a graphing utilit.. Use the reduced row-echelon form feature of a graphing utilit.. Solution:,. Solution:,. z z z Solution:,,. z z z Solution:,,. Solution:,. Solution:,. z Solution:,,. z Solution:,,. R R R Solution:,

6 Section. Matrices and Sstems of Equations. R R R R Solution:,. R R R R R Solution:,. R R R R R Solution:,. R R R R R Solution:,. R R R R... R R.... The sstem is inconsistent and there is no solution. Solution:,

7 Chapter Matrices and Determinants. R R a Solution: a a, a where a is a real number. z z z R R R R R R R z z z z Solution:,,. z z R R R R R R R R R R R z z z Solution:,, z

8 Section. Matrices and Sstems of Equations. R z z z R R R R R R R z z z z Solution:,,. z z z z z R R R R R R R R R z Solution:,, R R. z z z R R R R R R z z z z Solution:,,

9 Chapter Matrices and Determinants. R R R R R z z z R R z z z R R Solution:,, z. R R R R R R R R z z Let z a. a a a a Solution: z z z R a, a, a where a is an real number.. z z z R R R R R R z a R R. R R R R z w z w Let w a and z b. b a b a b a b a Solution: z w z w b a, b a, b, a where a and b are real numbers a a Solution: a, a, a where a is a real number

10 Section. Matrices and Sstems of Equations. z w z w R R R R R R w a z a b b a Solution: b a, b, a, a where a and b are real numbers. R R R R R R R R R The sstem is inconsistent and there is no solution.. R R R R R R The sstem in inconsistent and there is no solution.. Use the reduced row-echelon form feature of a graphing utilit. Let z a. z z z z z a Solution:, a, a where a is an real number

11 Chapter Matrices and Determinants. Use the reduced row-echelon form feature of a graphing utilit. z z z z z z a a a Solution: a, a, where a is a real number. Use the reduced row-echelon form feature of a graphing utilit. z z z w w w w z w Solution:,,,. Use the reduced row-echelon form feature of a graphing utilit. z z z z w w w w z w w z Solution:,,,

12 Section. Matrices and Sstems of Equations. Use the reduced row-echelon form feature of a graphing utilit. z w z w z z z w Let z a. Then a and a. Solution: a, a, a, where a is a real number. z z w w z w w w w w a, z a, a, a Solution: a, a, a, a where a is a real number. (a) z z z Solution:,, Both sstems ield the same solution, namel,,. z z z Solution:,,. (a) z z z The sstems do not ield the same solution. z z

13 Chapter Matrices and Determinants. (a) z z z z z z Solution:,, The sstems do not ield the same solution. Solution:,,. (a) z z z z z z The sstems do not ield the same solution.. z z z R R R R R This is a matri in row-echelon form. The row-echelon form feature of a graphing utilit ields this form. There are infinitel man matrices in row-echelon form that correspond to the original sstem of equations. All such matrices will ield the same solution, namel,,.. I I I I I R I I I R R I I I I I R R R R I I I I I

14 Section. Matrices and Sstems of Equations. Sstem of equations: Thus, A, B, C. So, A A B C rref B A B C A B C A B C A B A C A B C A C.. Sstem of equations: A, B, C A B A B C A B C A B A C A B C A B A C A B C rref C. amount at % amount at %, z amount at % z z...r R R R R R R R z z z z.z z..,,,,,.,,,,,,,,,,,,..,,,,,,,,,,, Solution: $, at %, $, at %, and $, at %,,,,,,,,,,,,,,

15 Chapter Matrices and Determinants. amount at %, amount at %,. a b c z amount at % a b c z, a b c a b c...z,..r R.R R R R R R R R.. R z, z, z,..,,,,,,.... Solution:,,,,,,,,,,,, Answer: $, at %, $, at %, $, at %,,,,,,,,, R R R R R R R a b b c c c c b b a a Equation of parabola:. f a b c f a b c f a b c f a b c R R R R R R R a b b c c c c b b a a Equation of parabola:

16 Section. Matrices and Sstems of Equations. (a),.,,.,,. a b c c a b c a b c R R a b b a Equation of parabola: R R (c) The maimum height is approimatel feet and the ball strikes the ground at approimatel feet. (d) The maimum occurs at the verte. b a..... feet..... The ball strikes the ground when... a b. a b. a.... B the Quadratic Formula and using the positive value for we have. feet. (e) The values found in part (d) are more accurate, but still ver close to the estimates found in part (c)..... (a) f at bt c f a b c. f a b c. f a b c. a b c a b c a b c a b c c b. b. a.. Equation of parabola: R R R R R R R R b c c a..t.t. (c) For, t..... When compared to the actual value of., this is not ver accurate. (d) For, t..... The model estimates that in,. million people will participate in snowboarding. This indicates that the number of participants will almost triple in ears which is probabl not a reasonable estimate.

17 Chapter Matrices and Determinants. (a) R R R R R R R R R R R R R R R R R Let t and s, then t, s t s t, s, t t, s s. Solution: s, t, s, s t, t, s, t s, t :,,,,,, (c) s, t :,,,,,,

18 Section. Matrices and Sstems of Equations. (a) R R R R R R R R Let t. t t Let s. s t s t s t s t Solution: s t, s t, s, t, t, where s and t are real numbers. When and, s t t t.,,,, (c) When and, s t t t.,,,,. False. It is a matri.. False. The rows are in the wrong order. To change this matri to reduced row-echelon form, interchange Row and Row, and interchange Row and Row.. False. Gaussian elimination reduces a matri until a rowechelon form is obtained and Gauss-Jordan elimination reduces a matri until a reduced row-echelon form is obtained.. z a a a One possible sstem is: z z z a a a a a a (Note that the coefficients of,, and z have been chosen so that the a-terms cancel.) a a a or z z z

19 Chapter Matrices and Determinants. (a) In the row-echelon form of an augmented matri that corresponds to an inconsistent sstem of linear equations, there eists a row consisting of all zeros ecept for the entr in the last column. In the row-echelon form of an augmented matri that corresponds to a sstem with an infinite number of solutions, there are fewer rows with nonzero entries than there are variables and no row has the first non-zero value in the last column... Interchange two rows.. The are the same.. Multipl a row b a nonzero constant.. Add a multiple of one row to another row.. A matri in row-echelon form is in reduced row-echelon form if ever column that has a leading has zeros in ever position above and below its leading.. f, f.. undef. undef. Vertical asmptote: Horizontal asmptote: Intercept:,. f The graph has a vertical asmptote at and a discontinuit at. Since the degrees of the numerator and the denominator are the same, there is a horizontal asmptote at.. f. g f Horizontal asmptote: Intercept:,

20 Section. Operations with Matrices. h ln. h.... Vertical asmptote: Intercept:,. f ln ln e.... Section. Operations with Matrices A B if and onl if the have the same order and a ij b ij. You should be able to perform the operations of matri addition, scalar multiplication, and matri multiplication. Some properties of matri addition and scalar multiplication are: (a) A B B A A B C A B C (c) cda cda (d) A A (e) ca B ca cb (f) c da ca da You should remember that AB BA in general. Some properties of matri multiplication are: (a) ABC ABC AB C AB AC (c) A BC AC BC (d) cab cab AcB You should know that I n, the identit matri of order n, is an n n matri consisting of ones on its main diagonal and zeros elsewhere. If A is an n n matri, then AI n I n A A. Vocabular Check. equal. scalars. zero; O. identit. (a) (iii) (iv) (c) (i). (a) (ii) (iv) (c) (i) (d) (iii) (d) (v) (e) (ii)

21 Chapter Matrices and Determinants.,.,.,,.,. (a) (c). (a) A B A B A (d) A B (c) A B A B A (d) A B. A, B (a) A B A B (c) (d) A B A. (a) (c) A B A B A (d) A B

22 Section. Operations with Matrices. A, B (a) A B A B (c) A (d) A B. (a) (c) (d) A B A B A A B

23 Chapter Matrices and Determinants. A, B (a) A B is not possible. A and B do not have the same order. (c) (d) A B is not possible. A and B do not have the same order. A A B is not possible. A and B do not have the same order.. (a) A B is not possible. A and B do not have the same order. A B is not possible. A and B do not have the same order. (c) A (d) A B is not possible. A and B do not have the same order

24 Section. Operations with Matrices X. X A B X A B. X A B. A B X X A B. A is and B is. AB is not possible.. A is, B is. AB is not possible.. A is, B is AB is.. A is, B is AB is. AB

25 Chapter Matrices and Determinants. A is, B is AB is.. A is, B is AB is. AB. A is, B is AB is.. A is, B is AB is..... A is, B is. AB is not possible.. A is and B is AB is not possible... (a) AB BA (c) A

26 Section. Operations with Matrices. (a) AB BA (c) A. (a) AB BA (c) A. (a) AB BA (c) A. (a) (c) AB BA A is not possible.. (a) AB BA (c) The number of columns of A does not equal the number of rows of A; the multiplication is not possible...

27 Chapter Matrices and Determinants... (a) R R R R X. (a) R R R R R R R R X. (a) R R R R R R X. (a) R R X R R R R R

28 Section. Operations with Matrices. (a). (a) A R R R R R R R R R R R R X R R R R R R R R R R R R X. (a). (a) R R R R R R R R R R R R R R R X X R R R R R R R R R R R R

29 Chapter Matrices and Determinants..... (a) Farmer s Fruit Fruit Market Stand Farm Each entr represents the number of bushels of each tpe of crop that are shipped to each outlet. B.. (c) A Each entr represents the profit per bushel for each tpe of crop. BA.. Apples Peaches $. $. $. The entries in the matri represent the profits for both crops at each of the three outlets.. BA $. $. $.,,,,, $,, $, The entries represent the costs of the three models of the product at the two warehouses.. ST $, $, $, $, $, $, The entries represent the wholesale and retail inventor values of the inventories at the three outlets.. P The P matri gives the proportion of the voting population that changed parties or remained loal to their part from the first election to the third.. P P P... P P P... P P P... P CONTINUED

30 Section. Operations with Matrices. CONTINUED As P is raised to higher and higher powers, the resulting matrices appear to be approaching the matri... P... P ST This represents the labor cost for each boat size at each plant. $. $. $. $. $. $.. (a) AB... Sales..... Profit Frida Saturda Sunda The entries in Column represent the total sales of the three kinds of milk for Frida, Saturda, and Sunda. The entries in Column represent each das total profit. Total profit for the weekend: $. (a) AB Sales ($) Profit.. The first column of AB gives the amount of sales for each octane. The second column gives the profit made b each octane. The store s profit for the weekend is $ $. $. $... (a) B Biccled [ Jogged. Walked ] -minute time periods BA. -pound person [. -pound person.] Calories burned The first entr represents the total calories burned b the -pound person and the second entr represents the total calories burned b the -pound person.. (a) Individual Famil costs costs A... B... CONTINUED... Comprehensive plan HMO standard plan HMO plus plan... Comprehensive plan HMO standard plan HMO plus plan

31 Chapter Matrices and Determinants. CONTINUED Change in Change in individual famil costs cost Comprehensive plan HMO standard plan HMO plus plan Emploees choosing the comprehensive plan have a decrease in cost while those choosing the other two have an increased cost. (c) Dividing each entr of matri A b ields A......, (d) If the costs increase b % net ear, then the new cost matri would be: A.A... Monthl individual cost A.A B Monthl famil cost... Comprehensive plan HMO standard plan HMO plus plan. True. The sum of two matrices of different orders is undefined.. False. For most matrices, AB BA. For, A is of order, B is of order, C is of order and D is of order.. A C is not possible. A and C are not of the same order.. B C is not possible. B and C are not of the same order.. AB is not possible. The number of columns of A does not equal the number of rows of B.. BC is possible. The resulting order is.. BC D is possible. The resulting order is.. CB D is not possible. The order of CB is, but the order of D is.. DA B is possible. The resulting order is.. BC DA is possible. The resulting order is.. AC BC Thus, AC BC even though A B.. AB AB O and neither A nor B is O.. The product of two diagonal matrices of the same order is a diagonal matri whose entries are the products of the corresponding diagonal entries of A and B.

32 Section. Operations with Matrices. (a) A i A A A A A A i B i B i i i i i i i i i i i ii i i i i ii i i ii i i i i ii i i i i i i i ii i i ii and i i and i i and i I, the identit matri.. or Solutions:, or Solutions:,.. ± ± ± Solutions:, or ± b the Quadratic Formula Solutions:,,.. or Solutions:, ± i ± ± i ± ± Solutions:, ±. Eq. Eq. Eq. Add equations. Solution:, (, )

33 Chapter Matrices and Determinants. Equation Equation Eq. Eq. Add equations. (, ) Solution:,. Equation Equation Eq. Add equations. (, ) Solution:,. Equation Equation Eq. Eq. Add equations. (, ) Solution:, Section. The Inverse of a Square Matri You should know that the inverse of an n n matri A is the n n matri A, if is eists, such that AA A A I, where I is the n n identit matri. You should be able to find the inverse, if it eists, of a square matri. (a) Write the n n matri that consists of the given matri A on the left and the n n identit matri I on the right to obtain A I. Note that we separate the matrices A and I b a dotted line. We call this process adjoining the matrices A and I. If possible, row reduce A to I using elementar row operations on the entire matri A I. The result will be the matri I A. If this is not possible, then A is not invertible. (c) Check our work b multipling to see that AA I A A. a b The inverse of is A if ad cb. ad bc d b A c d c a You should be able to use inverse matrices to solve sstems of linear equations if the coefficient matri is square and invertible. Vocabular Check. square. inverse. nonsingular; singular. A B

34 Section. The Inverse of a Square Matri. AB BA. AB BA. AB BA.. AB BA AB BA.. AB BA AB CONTINUED

35 Chapter Matrices and Determinants. CONTINUED BA. AB BA. AB AB BA

36 . AB BA Section. The Inverse of a Square Matri. A I R R A I A. A I R R R R A I A. A I R R R R A I A. A I R R R R R R I A A

37 Chapter Matrices and Determinants. A I R R R R A. I... A A I R R R R R R A I A. A I R R The two zeros in the second row impl that the inverse does not eist.. A I A R R R R R R R I A. A A has no inverse because it is not square.. A A has no inverse because it is not square.. A I R R R R R R R R R R R R R A R I A

38 Section. The Inverse of a Square Matri. A I R R R R R R R R R R R R A I A.. A I R R R R R R A R R A I I A R R R R Since the first three entries of row are all zeros, the inverse of A does not eist.. A I R R R A I A

39 Chapter Matrices and Determinants. A I R R R R R R R R R R R R R R R I A A. A A. A A. A A..... A A.. A A. A does not eist.

40 .. A.. A Section. The Inverse of a Square Matri A... A A A does not eist.. A A. A A. A A. A a c A ad bc A b d, A ad bc d c b a. A ad bc A. A. ad bc Since ad bc, A does not eist. A ad bc A. A A. ad bc A ad bc A

41 Chapter Matrices and Determinants. Solution:,. Solution:,. Solution:,. Solution:,. z Solution:,,. z Solution:,,. Solution:,,,. w z Solution:,,,. A. A A Solution:, A Solution:,. A A does not eist. A... This implies that there is no unique solution; that is, either the sstem is inconsistent or there are infinitel man solutions. Find the reduced row-echelon form of the matri corresponding to the sstem..r R R The given sstem is inconsistent and there is no solution.. A. A Solution:,

42 Section. The Inverse of a Square Matri. A A Solution:,. A A Solution:,. A Find A. R R R A I R R R R R R R R R R R R R R R R A z Solution:,, I A

43 Chapter Matrices and Determinants. A z A Solution:,,. A A does not eist. This implies that there is no unique solution; that is, either the sstem is inconsistent or the sstem has infinitel man solutions. Use a graphing utilit to find the reduced row-echelon form of the matri corresponding to the sstem. z z Let Then z a. a and a. Solution: a, a, a where a is a real number. A A does not eist. This implies that there is no unique solution; that is, either the sstem is inconsistent or the sstem has infinitel man solutions. Use a graphing utilit to find the reduced row-echelon form of the matri corresponding to the sstem. z z. Let z a. Then a and a. Solution: a, a, a where a is a real number A. A..... z Solution:,, A A..... z. Solution:,, A A w z Solution:,,,

44 Section. The Inverse of a Square Matri.. A A w... z Solution:.,.,.,. A. R R R R R R. A I R R R R R R X A B...., I A Solution: $ in AAA-rated bonds, $ in A-rated bonds, $ in B-rated bonds

45 Chapter Matrices and Determinants. A. A I R R R R R R R R.. R R R R X A B..., I A Solution: $ in AAA-rated bonds, $ in A-rated bonds, $ in B-rated bonds.. Use the inverse matri from Eercise. X A B A, Solution: $ in AAA-rated bonds, $ in A-rated bonds, $ in B-rated bonds. Use the inverse matri A from Eercise. X A B,,,,, Solution: $, in AAA-rated bonds, $, in A-rated bonds, and $, in B-rated bonds.

46 Section. The Inverse of a Square Matri. (a) A I I A A I R R R R R R R R I R R R R R Solution: I amperes, I amperes, I amperes I A I I I Solution: I amperes, I amperes, I amperes. (a) (e) n ; n i ; n i....; i n i i n i i... i Sstem: i b a. b a.t..t. t. Since t represents, the model projects that the number of licensed drivers will reach million during. b., a. ;..... The least squares regression line is.t.. (c) For, t ;.... This projects about million licensed drivers in. (d) The projected value is ver close to the actual value.. True. If B is the inverse of A, then AB I BA.. True. If A and B are both square matrices and AB I n, it can be shown that BA I n.

47 Chapter Matrices and Determinants. AA a c A A b d ad bc ad bc ad bc ad bc d c ad bc d c b a a c b a ad bc a c b d d c b a b d ad bc ad bc ad bc. (a) Given A a Given A a a, A a a, A a. a a a. a In general, the inverse of a matri in the form of A is... a a a a nn.. or <. < < ln ln or < < < < ln ln ln ln.. e. log.. ln ln e log... ln e ln e ln e ln ln ln. ± ± Choose the positive value onl:.. Answers will var.

48 Section. The Determinant of a Square Matri Section. The Determinant of a Square Matri You should be able to determine the determinant of a matri of order b using the difference of the products of the diagonals. You should be able to use epansion b cofactors to find the determinant of a matri of order or greater. The determinant of a triangular matri equals the product of the entries on the main diagonal. Vocabular Check. determinant. minor. cofactor. epanding b cofactors

49 Chapter Matrices and Determinants.. (a) M M M C M C M C M (a) M M M C M C M C M M C M M C M.. (a) M C M (a) M C M M C M M C M M C M M C M M C M M C M.. (a) M M M M M M M M M C M C M C M C M C M C M C M C M C M (a) M M M M M M M M M C M C M C M C M C M C M C M C M C M

50 Section. The Determinant of a Square Matri.. (a) M M M M M M M M M C M C M C M C M C M C M C M C M C M (a) M M M M M M M M M C M C M C M C M C M C M C M C M C M. (a). (a)

51 Chapter Matrices and Determinants. (a). (a). (a). (a). Epand along Column.. Epand along Row.

52 Section. The Determinant of a Square Matri. Epand along Row.. Epand along Column.. (Upper triangular). Epand along Row.. Epand along Column.. Epand along Row.. (Upper triangular). Epand along Row.. Epand along Column.. Epand along Row.. Epand along Column.

53 Chapter Matrices and Determinants. Epand along Row.. Epand along Column, then along Column.. Epand along Column (a) (c). (a) (c) A B AB (d) (d) AB

54 Section. The Determinant of a Square Matri. (a) (c) (d). (a) (c) A B AB (d) AB. (a) (c) (d). (a) (c) A B AB (d) AB. (a) (c) (d). w w Thus, w z z. (a) wz. w wz wz c w z w z. (c) A B AB (d) AB So, w c cz c w c cwz c cwz cz z cwz z.

55 Chapter Matrices and Determinants. w.. w wz. cw wz c cw wz z c Thus, w a b a a z z z w cw z c. z z z z z z z z z a a b a z z z z z z z z z z z z z z z z zz z z z a b a a b a w cw So, w cw cw cw c. c b a a a b a b a a a b a a a a a b a a a a b a a a ba b a aaa b a aa aa b a b a a b a a b a a a a b.. a a b ab b a a b a ab b b a b. or or or. or

56 Section. The Determinant of a Square Matri. u uv v.. e e e e e e e. e e. ln e e e e e e e e ln. ln ln ln ln ln ln. True. If an entire row is zero, then each cofactor in the epansion is multiplied b zero.. True. If a square matri has two columns that are equal, then elementar column operations can be used to create a column with all zeros.. Let A B, A B Thus, Your answer ma differ, depending on how ou choose A and B. A A and B, B A B A B.., A B. (a) For an n n matri n > with consecutive integer entries, the determinant appears to be.

57 Chapter Matrices and Determinants. A square matri is a square arra of numbers. The determinant of a square matri is a real number.. Let A A and A. A A So, A A.. (a) Column and Column were interchanged. Row and Row were interchanged.. (a) Multipling Row of the matri b and adding it to Row gives the matri Multipling Row of the matri b and adding it to Row gives the matri. (a) A B A, B Row was multiplied b. B A A B A, B.. Column was multiplied b and Column was multiplied b. B A A

58 Section. The Determinant of a Square Matri. (a) A A (c) A Using cofactors and a, C A C, A, A A C C C C. In each case, the determinant of the matri is the product of the diagonal entries. From this, one would conjecture that the determinant of a diagonal matri is the product of the diagonal entries.. f. g Since f is a polnomial, the domain is all real numbers. An odd root of a number is defined for all real numbers. Domain: all real numbers. h. Critical numbers: ± Test intervals:,,,,, Test: Is? Solution:, Domain of h: A ± Domain: all real numbers ±. gt lnt. t > t > Domain: all real numbers t > fs e.s The eponential function Ae is defined for all real numbers. Domain: all real numbers. <.

59 Chapter Matrices and Determinants. A I R R R R R A. I A A I R R R R R R R R A I A. A I R R R R R R R R The zeros in Row impl that the inverse does not eist.. A I R R R R R R R R R CONTINUED

60 Section. Applications of Matrices and Determinants. CONTINUED R R R R R R R R A R I A Section. Applications of Matrices and Determinants You should be able to use Cramer s Rule to solve a sstem of linear equations. Now ou should be able to solve a sstem of linear equations b graphing, substitution, elimination, elementar row operations on an augmented matri, using the inverse matri, or Cramer s Rule. You should be able to find the area of a triangle with vertices,,,, and,. Area ± The ± smbol indicates that the appropriate sign should be chosen so that the area is positive. You should be able to test to see if three points, are collinear.,,,, and,,, if and onl if the are collinear. You should be able to find the equation of the line through b evaluating., and, You should be able to encode and decode messages b using an invertible n n matri. Vocabular Check. Cramer s Rule. colinear. A ±. crptogram. uncoded; coded

61 Chapter Matrices and Determinants. Solution:,. Solution:,. Since Cramer s Rule does not appl., The sstem is inconsistent in this case and has no solution.. Solution:, Solution:, Solution:, z z z, D,, z Solution:,,

62 Section. Applications of Matrices and Determinants. D z z z z. z z z z Solution:,,, D Solution:,,. D z z z z. z z z z, Solution:,, D Solution:,,. z z z, D,, z Solution:,,

63 Chapter Matrices and Determinants. z z z Solution:,, D,, z. D z z z. Vertices: Area,,,,, square units Cramer s Rule does not appl.. Vertices: Area,,,,, square units. Vertices: Area,,,,, square units. Vertices:,,,,, Area square units. Vertices: Area,,,,, square units. Vertices: Area,,,,, square units. Vertices:,,,,, Area square units

64 Section. Applications of Matrices and Determinants. Vertices:,,,,, Area square units. Vertices:,,,,, Area square units. Vertices: Area,,,,, square units. ± ± ±. ± ± ± ± ± ± or ± ± or. ±. ± ± ± ± ± or ± ± ± ± ± or ±. Vertices:,,,,, Area square miles. Vertices:,,,,, Area square feet

65 Chapter Matrices and Determinants. Points:,,,,, The points are collinear.. Points:,,,,, The points are not collinear.. Points:,,,,, The points are not collinear.. Points:,,,,, The points are collinear.. Points:,,,.,,... The points are collinear.... Points:,,,.,,.. The points are not collinear.....

66 Section. Applications of Matrices and Determinants. Points: Equation:,,,. Points:,,, Equation: or. Points:,,, Equation:. Points: Equation:,,, or. Points: Equation:,,,. Points: Equation:,,, or. The uncoded row matrices are the rows of the matri on the left. T U E N I R I R B V T O L I R E C Y Solution:

67 Chapter Matrices and Determinants. Solution: Uncoded matrices:,,,,, Encoded matrices:,,,,, Encoded message: In Eercises, use the matri A [ ].. C [ A L L ] [ A T ] [ A A A A N O ] [ Crptogram: O N ]. I C E B E R G D E A D A H E A D CONTINUED

68 Section. Applications of Matrices and Determinants. CONTINUED Crptogram:. H [ A P P ] [ Y B ] [ I R T ] [ A A A A A H D A ] [ Crptogram: Y ]. O P E R A T I O N _ O V E R L O A D Crptogram:

69 Chapter Matrices and Determinants. A H P Y N W Y A A P E E R Message: HAPPY NEW YEAR. A B R O N C O S W I Message: BRONCOS WIN SUPER BOWL N S U P E R B O W L. A C S I C C E L S S A E D A N L Message: CLASS IS CANCELED

70 Section. Applications of Matrices and Determinants. A H A V E A G R E A T W E E K E N D Message: HAVE A GREAT WEEKEND. A S D L E E A S N P N Message: SEND PLANES. R E T U R N AT DA W N Message: RETURN AT DAWN

71 Chapter Matrices and Determinants. Let A be the matri needed to decode the message. A A O R N M E E T N G T O E T M O I H R N Message: MEET ME TONIGHT RON. (a) (c) n ; n i ; n i ; n i ; n i ; n i, i i n i i, i n i i, i Sstem: D c b a The least squares regression parabola is.t.t.,,,, i c b a, c b a, c b a, i,,,,.,,,. i, (d) The intersection of the regression parabola and the line, is about t., so the number of cases waiting to be tried will reach, in about.

72 Section. Applications of Matrices and Determinants. False. In Cramer s Rule, the denominator is the determinant of the coefficient matri.. True. If the determinant of the coefficient matri is zero, the solution of the sstem would result in division b zero which is undefined.. False. If the determinant of the coefficient matri is zero, the sstem has either no solution or infinitel man solutions.. Answers will var. To solve a sstem of linear equations ou can use graphing, substitution, elimination, elementar row operations on an augmented matri (Gaussian elimination with back substitution or Gauss-Jordan elimination), the inverse of a matri, or Cramer s Rule.. Solution:, Equation Equation Eq. Add equations.. Equation Equation Eq. Eq. Add equations. Solution:,. z z z A z A Solution:,,. A z A z z z Solution:,,. Objective function: Constraints: z At, : z At, : z At, : z At, : z (, ) (, ) The minimum value of occurs at,. The maimum value of occurs at,. (, ) (, (. Objective function: z Constraints: Since the region is unbounded, there is no maimum value of the objective function. To find the minimum value, check the vertices. At, : z At, : z At, : z ( The minimum value of occurs at,. (, ) (, ) (, )

73 Chapter Matrices and Determinants Review Eercises for Chapter.. Order:. Since the matri has two rows and four columns, its order is. Order:.. Since the matri has one row and five columns, its order is..... z z z. z z. z w z w z w. R R R R R R R. R R R R R R R R R. Solution:,, z z z. z z z Solution:,,

74 Review Eercises for Chapter. Solution:,, z z z. z z Solution:,,. R R R R R Solution:,. R R R R Solution:,. R R R R R..... R R..... Solution:.,., R..... R R R Solution:.,.,......

75 Chapter Matrices and Determinants.. R R R R R R R R R Solution:,, R R R R R Let z a, then: z R R R R a a.. R R R R R R R R R R R z z z z Solution:,, R R R R R R Because the last row consists of all zeros ecept for the last entr, the sstem is inconsistent and there is no solution. a a Solution: a, a, a where a is an real number

76 Review Eercises for Chapter. R R R R R R R R R R R R R R R w z z Solution:,,,. R R R R R R R Because the last row consists of all zeros ecept for the last entr, the sstem is inconsistent and there is no solution.. R R R R R R R R R R R R R R R,, z Solution:,,

77 Chapter Matrices and Determinants. R R R R R R R R z R R R R z z z R R R Solution:,,. R R R R R R R R R R R R R R R R R R,, z Solution:,,. R R z z z R R R R R R R R R R R R R R R,, z Solution:,,

78 Review Eercises for Chapter. Use the reduced row-echelon form feature of a graphing utilit.,, z, w Solution:,,,. Use the reduced row-echelon form feature of the graphing utilit. The sstem is inconsistent and there is no solution... and and..,,. (a) (c) A B A B A (d) A B. (a) A B A B (c) A (d) A B

79 Chapter Matrices and Determinants. (a) A B A B (c) A (d) A B. (a) A B is not possible. A and B do not have the same order. A B is not possible. A and B do not have the same order. (c) A (d) A B is not possible. A and B do not have the same order... Since the matrices are not of the same order, the operation cannot be performed X A B

80 ... X A B X B A X A B Review Eercises for Chapter. A and B are both so AB eists. AB. Not possible because the number of columns of A does not equal the number of rows of B.. Since A is and B is, AB eists. AB. AB.. Not possible because the number of columns of the first matri does not equal the number of rows of the second matri.

81 Chapter Matrices and Determinants A...A.

82 Review Eercises for Chapter. BA... $, $, The merchandise shipped to warehouse is worth $,, and the merchandise shipped to warehouse is worth $,.. (a) T TC Your cost with compan A is $.. Your cost with compan B is $... AB I BA I. AB I BA I. AB I BA I

83 Chapter Matrices and Determinants.. AB BA A A I R R R R R R I A I I. A I R R R R R R A I A. A I R R R R R R R R R R R R R R A I A

84 Review Eercises for Chapter. R R R R A I R R R R R R R R R R R R A I A.. A A does not eist A A.... A A. A ad bc A

85 Chapter Matrices and Determinants. A A. A ad bc A.. Solution:, Solution:,.. Solution:, Solution:,. z z z z Solution:,,. z z z z Solution:,,. z z z z Solution:,,

86 Review Eercises for Chapter. z z z z Solution:,,. Solution:,., Solution:, )..... z z z z Solution:,,. z z z z,, z Solution:,,....

87 Chapter Matrices and Determinants. (a) M M M M C M C M C M C M. (a) M C M M M C C C M M M M.. (a) M (a) M M M M M M M M M M M M M M M M M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M. Epand using Column.

88 Review Eercises for Chapter. Epand using Row.. Epand along Row.. Epand using Row, then use Row of each matri.. Solution:,,. Solution:,,

89 Chapter Matrices and Determinants. D z z z z Solution:,, (). z z z Solution:,,, D,, z.,,,,, Area square units

90 Review Eercises for Chapter.,,,,,.,,,,, Area square units Area square units.,,,,,. Area square units,,,,, The points are collinear.. Points:,,,,, The points are collinear..,,,.,,,.,,,..,.,., Multipl both sides b.

91 Chapter Matrices and Determinants. L [ O A O ] Crptogram: K [ O ] U [ T ] B [ E L ] O [ W ]. R E T U R N T B A S E A A A A A A Crptogram:

92 Review Eercises for Chapter. A Message: SEE YOU FRIDAY U F S E E Y O R I D A Y. A M E O E E I U A T R T Y Y H F C B W H O Message: MAY THE FORCE BE WITH YOU. False. The matri must be square.. True. Epand along Row. a a a a a a a c a c a c a c a a a a a c a a a a a c a a a a a a a a a a a a a a a c a a a a a a a a a a a a a a Note: Epand each of these matrices along Row c a a a a a c c c to see the previous step. a a a a a a a a c a a a a

93 Chapter Matrices and Determinants. The matri must be square and its determinant nonzero to have an inverse.. If A is a square matri, the cofactor C ij of the entr a ij is ij M ij, where M ij is the determinant obtained b deleting the ith row and jth column of A. The determinant of A is the sum of the entries of an row or column of A multiplied b their respective cofactors.. No. Each matri is in row-echelon form, but the third matri cannot be achieved from the first or second matri with elementar row operations. Also, the first two matrices describe a sstem of equations with one solution. The third matri describes a sstem with infinitel man solutions.. The part of the matri corresponding to the coefficients of the sstem reduces to a matri in which the number of rows with nonzero entries is the same as the number of variables.. ± ± Problem Solving for Chapter. A (a) AT T AAT Original Triangle AT Triangle AAT Triangle The transformation A interchanges the and coordinates and then takes the negative of the coordinate. A represents a counterclockwise rotation b. A AAT A AAT IAT AT A AT A AT IT T A (, ) (, ) A (, ) represents a clockwise rotation b. (, ) (, ) (, ) (, ) (, ) (, )

94 Problem Solving for Chapter. (a) +.%.%.% Northeast.%.%.% Midwest.%.%.% South.%.%.% Mountain.%.%.% Pacific +.%.%.% Northeast.%.%.% Midwest.%.%.% South.%.%.% Mountain.%.%.% Pacific Change in Percent of Population from to +.%.%.%.%.%.%.%.%.%.%.%.%.%.%.% Northeast Midwest South Mountain Pacific (c) All regions show growth in the + age bracket, especiall the South. The South, Mountain and Pacific regions show growth in the age bracket. Onl the Pacific region shows growth in the age bracket.. (a) A Aisidempotent. A (c) A A is not idempotent. A A A (d) A A A is not idempotent. A is not idempotent.. A (a) A A I A I A Thus, A I A. (c) A A I A A I A IA I A IA I I AA I Thus, A I A.

95 Chapter Matrices and Determinants. (a) (c) ,.,.,,,, Gold Cable Compan:, households Gala Cable Compan:, households Nonsubscribers:, households,,,., ,., Gold Cable Compan:, households Gala Cable Compan:, households Nonsubscribers:, households ,.,., Gold Cable Compan:, households Gala Cable Compan:, households Nonsubscribers:, households,,, (d) Both cable companies are increasing the number of subscribers, while the number of nonsubscribers is decreasing each ear.. A A If A A, then. Equating the first entr in Row ields. Now check in the other entries: Thus,.. If A is singular then ad bc. Thus,.. From Eercise we have the singular matri A Also, A where A A. has this propert.. a a Thus, b b a a c c b b a bb cc a a b a c ab ac b c bc b b c c c c a a c c a a a bb cc a. b b bc b c ac a c ab a b

96 Problem Solving for Chapter. a bb cc aa b c a b a c ab ac b c bc a b c b c b a b c c a c a c a b a b bc b c ac a c ab a b Thus, a b c a bb cc aa b c. a b c a c b. b a b c a b c a c a d a c c. b d a b b c d From Eercise a b c d. S N S F D S N F Element Atomic mass Sulfur Nitrogen Fluoride N F. Let cost of a transformer, cost per foot of wire, z cost of a light. z z z B using the matri capabilities of a graphing calculator to reduce the augmented matri to reduced row-echelon form, we have the following costs: Transformer $. Foot of wire $. Light $. rref.

97 Chapter Matrices and Determinants. A A T AB B T A T,, Thus, AB T B T A T., B B T AB T. A A R E M E M B E R S E P T E M B E R T H E E L E V E N T H REMEMBER SEPTEMBER THE ELEVENTH

98 Problem Solving for Chapter. (a) w w w z w z z z w w, w JOHN RETURN TO BASE J H E U N T A E O N R T R O B S z, z z A. A A A and A Conjecture: A A. Let then A, Let A Let A, A then then Conjecture: If A is an n n matri, each of whose rows add up to zero, then. A., A. A.. (a) Answers will var. (c) A A so A n for n an integer. B, B so B n for n an integer. A if A is. B (d) Conjecture: If A is n n, then A n.

99 Chapter Matrices and Determinants Chapter Practice Test. Put the matri in reduced row-echelon form. For Eercises, use matrices to solve the sstem of equations.... z z z. Multipl.. Given A find A B. and B,. Find fa. f, A. True or false: A BA B A AB B where A and B are matrices. (Assume that A, AB, and B eist.) For Eercises, find the inverse of the matri, if it eists.... Use an inverse matri to solve the sstems. (a) For Eercises, find the determinant of the matri....

100 Practice Test for Chapter. Evaluate.. Use a determinant to find the area of the triangle with vertices,,,, and,.. Find the equation of the line through, and,. For Eercises, use Cramer s Rule to find the indicated value.. Find.. Find z. z z. Find