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Unformatted text preview: I 9.1.1 In Problems [—4, solve each linear system of equations. In addition,
for each system, graph the two lines corresponding to the two
equations in a single coordinate system and use your graph to
explain your solution. 1. x— y: 1 2. 2x+3y= 6
x—2y=—2 x—4y=—4
3.x—3y=6 4.2x+ y=§
y=3+31x 6x+3y=1 5. Determine e such that . 2x—3y=5 4x—6y=c has (a) inﬁnitely many solutions and (b) no solutions. (c) Is it
possible to choose a number for c so that the system has exactly
one solution? Explain your answer. 6. (3) Determine the solution of —2x+3y=5
ax— y=1 in terms of a. (b) For which values of a are there no solutions, exactly one
solution. and inﬁnitely many solutions? 7. Show that the solution of anxi + “124‘: = [71
azixi + azzxz = b2 is given by
azzbi  a12b2
X1 =
011022  0211112
and
“02151 + aubz
X2 = anazz  021012
8. Assume that the system auxl + 012162 = b1
021161 + azzxz = b2 has inﬁnitely many solutions. Determine the number of solutions
if you change
(8) an (b) b. In Problems 9—16, reduce the system of linear equations to upper
triangular form and solve. ' 9. 2x— y=3 10.5x—3y=2
x~3y=7 2x+7y=3 11. 7x— y=4 12. 5x+2y=8
3x+2y=1 ~x+3y=9 13. 3x—y=1 l4.2x+3y=5
—3x+y=4 — y=—2+§x 15. .t+2y=3 16. x—Zy: 2
4y+2x=6 4y—2x=—4 9.1 I Linear Systems 443 Section 9.1 Problems 17. Zach wants to buy ﬁsh and plants for his aquarium. Each ﬁsh
costs $2.30; each plant costs $1.70. He buys a total of 11 items and
spends a total of $21.70. Set up a system of linear equations that will allow you to determine how many ﬁsh and how many plants
Zach bought, and solve the system. 18. Laboratory mice are fed with a mixture of two foods that
contain two essential nutrients. Food 1 contains 3 units of nutrient A and 2 units of nutrient B per ounce; food 2 contains 4 units of
nutrient A and 5 units of nutrient B per ounce. (a) In what proportion should you mix the food if the mice
require nutrients A and B in equal amounts? (b) Assume now that the mice require nutrients A and B in the ratio 1:2. Is it possible to satisfy their dietary needs with the two
foods available? 19. Show that if
allaZZ  (man 95 0
then the system
aux; + aux; = 0
azixi + a22x2 = 0 has exactly one solution, namely, x1 = 0 and x2 = 0. l 9.1.2 In Problems 20—24, solve each system of linear equations
20.2x—3yl z=—1 21. Sx—y+2z= 6
x+y—22=—3 x+2y— z=—1
3x—2y+ z: 2" 3x+2y—2z= 1 22. x+4y—3z=—13 23. ——2x+4y z=—1
2x—3y+5z= 18 x+7y+2z=—4
3x+ y—2z= 1 3x—2y+3z=—3 24. 2x— y+3z=3
2x + y + 42 = 4
2x  3y + 22 = 2
In Problems 25—28, ﬁnd the augmented matrix and use it to solve
the system of linear equations.
25. —x ——2y+3z=—9
2x + y  z = 5
4x —— 3y + Sz = —9 26.3x—2y+ z: 4
4x+ y—2z=—12
2x—3y+ z: 7 27y+x=3 28. 2x—z=1
z—y: —1 y+3z=x—1
x+z=2 x+ z=y—3 In Problems 2934, determine Whether each system is overdeter
mined or underdetermined; then solve each system. 29. x—2y+z=3 30.x—y =2
2x—3y+z= x+y+z=3
31.2x—y=3 32. 4y—3z=6
xy: 2y+ z='1
3x—y= y+ z=0
33.2x—7y+ z: 34. 3x+y=1
x+ y—22=4 x—y=0
4x :1 s_. <‘— ....1.:..;' .‘ s , p 4___ /
gnawW . . _ .. “.2": 22.; 444 Chapter 9 I Linear Algebra and Analytic Geometry 35. SplendidLawn sells three types of lawn fertilizer: SL 244—
8, SL 21—7—12 and SL 17—00. The three numbers refer to the
percentages of nitrogen, phosphate, and potassium. in that order,
of the contents (For instance, 100 g of SL 24—4—8 contains 24 g
of nitrogen.) Suppose that each year your lawn requires 500 g
of nitrogen. 100 g of phosphate. and 180 g of potassium per each day. Each individual of species 1 consumes 3 units of food A
and 5 units of food B. each individual of species 2 consumes 2
units of food A and 3 units of food 8, and individual of species 3
consumes 1 unit of food A and 2 units of food 8. Each day, 500
units of food A and 900 units of food 8 are supplied. How many
individuals of each species can be reared together? Is there more 1000 square feet. How much of each of the three types of fertilizer than one solution? What happens if we add 550 units of a third
should you apply per 1000 square feet per year? type of food. called C, and each individual of species 1 consumes 2 units of food C, each individual of species 2 consumes 4 units of
36. Three different species of insects are reared together in a food C, and each individual of species 3 consumes 1 unit of food laboratory cage. They are supplied with two different types of food C 7 I 9.2 Matrices We introduced matrices in the previous section; in this section, we will learn various
matrix operations. :I 9.2.1 Basic Matrix Operations Recall that an m x n matrix A is a rectangular array of numbers with m rows and n
columns. We write it as an 012 at»
021 £222 02 A: ’1 =[aij]lsi5m
................... l_jsn
am] and “mu We will also use the shorthand notation A = [aij] if the size of the matrix is clear.
We list a few simple deﬁnitions that do not need much explanation. ‘Deflr'titlon suppose that A =
Then“. _ _ ._ {any} and '36 = [by] are two mvx n matrices“ "A 4.. By; .
if‘and only if, for alll 5i 5m andl 5‘11" 5 n, air .=* bu. This definition says that we can compare matrices of the same size, and they are equal if all their corresponding entries are equal. The next deﬁnition shows how to
add matrices. Definition Suppose that A = [(2.7] and B ’= [by] are two m x n matrices.  :
Then ‘ ‘
C = A + B is an m x n matrix with entries Cij=alj+bij forlﬁiSM,ISjS" Note that addition is deﬁned only for matrices of equal size. Matrix addition satisﬁes
the following two properties: 1. A+B=B+A
2. (A+B)+C=A+(B+C) ‘S'ectiomezf Problemssf; 9.2 I Matrices 465 I 9.2.1, 9.2.2
In Problems 1—6, let ”[1. 3]. 142:]. c=ii If] 1. Find A —— B +20 2. Find —2A + SR.
3. Determine D so that A + B = 2A  B + D. 4. ShowthatA+B = B+A. 5. Showthat(A+B)+C=A+(B+C). 6. Show that 2(A + B) = 2A + 28» In Problems 7—12, let 13—1 5~14
A=241,B=201,
0—22 1—3~3
—204
C=1—31
002 7. Find2A+3B—C. a. Find3C~B+§A.
9. DetermineDsothatA+B+C+D =0. 10. Determine DsothatA+4B=2(A+B)+D.
. ShowthatA+B=B+A. . Showthat(A+B)+C=A+(B+C). . ShowthatifA+B=C,thenA=C—B. 11
12
13
14. Find the transpose of 15. Find the transpose of —1 o 3
A=[21—4] 16. Suppose A is a 2 x 2 matrix. Find conditions on the entries of
A such that A + A’ = 0
17. Suppose that A and B are m x n matrices. Show that (A+B)’=A’+B’
18. Suppose that A is an m x n matrix. Show that
(A')' = A 19. Suppose that A is an m x n matrix and k is a real number.
Show that (kA)/ = kA' 20. Suppose that A is an m x k matrix and B is a k x n matrix.
Show that (A B)’ = B'A'
In Problems 21—26, let .413], B=[_f it c=ié ii 21. Compute the following:
(a) AB (b) BA 22. Compute ABC. 23. Show that AC 75 CA. 24. Show that (AB)C = A(BC). 25. Show that (A + B)C = AC + BC.
26. Show that A(B + C) = AB + AC. 27. Suppose that A is a 3 x 4 matrix and B is a 4 x 2 matrix. What
is the size of the product AB? 29. Suppose that A is a 4 x 3 matrix. B is a 1 x 3 matrix, C is a 3 x1
matrix, and D is a 4 x 3 matrix. Which of the matrix multiplications that follow are deﬁned? Whenever it is deﬁned. state the size of
the resulting matrix. (a) BD’ (b) D’A (c) ACB 30. Suppose that A is anl x p matrix, B is an m x q. matrix, and
C is an n x r matrix. What can you say about I, m, n, p, q, and r if the products that follow are deﬁned? State the size of the resulting
matrix. (a) ABC
31. Let 1 3 1 2 0 1
(a) Compute AB.
32. Let (b) AB’C (c)BAC/ (d) A’CB’ (b) Compute B’A. —1
A=[14 —2] and B: 2
3 (b) Compute BA. 2 1
A  i —1 3]
Find A2. A3, and A4.
34. Suppose that (a) Compute AB.
33. Let Show that (A B)’ = B’A’. 35.Let
01
B[1 o] (a) Find 82, B3, 3‘, and B5. (b) What can you say about 8" when (i) k is even and (ii) k is
odd? 36. Let Show that l; = [32 = 133. 37. Let
1 3 I 0
A==[0 _2] and lz=[0 I] Show that AI; = 12A = A. hf 4‘1ﬁ'w *4 466 Chapter 9 u Linear Algebra and Analytic Geometry 38. Let 52. (a) Show that ifX = AX + D. then
1 3 0 1 0 0 X=(l—A)‘1D
A = fl) :; f and I“ = 3 (1, (1) provided that 1 — A is invertible. (b) Suppose that
Show that AI; = 13A = A. 3 2 D __ ~2
In Problems 39—42, write each system in matrix form. A = 0 _1 and _ 2
, 39: 2M +3i2  I: =0 40’ 2"? _ 31 = x] C l A " d 1 ' X
2x2 + x; =1 4x1 + x3 ___ 7x2 ompute( — ) .’ an use your resutin (a) to compute .
53. Use the determinant to determine whether the matrix
X] —2.X3=2 .l‘z—X1: X3
41. 2xi—3xz=4 42. x1—2x2+ x3=l A=[i 1]
—x1+ x2=3 —2x1+ x2—3x3=0
3x1 = 4 is invertible.
54. Use the determinant to determine whether the matrix
1 9.2.3
43. Show that the inverse of A __ [ —1 3]
" 0 3
A _ 2 1
_ 1 1 ' is invertible.
. 55. Use the determinant to determine whether the matrix
15
_ 1 —1 4 —1
B ‘ —1 2] A = [8 4] 44. Show that the inverse of l
A=l is invertible. 2 3 1 56. Use the determinant to determine whether the matrix
5 2 3 1 2
1 2 0 = ‘
A [1 2i
[S 6 2 7 is invertible.
“‘3 3 3 57. Suppose that
B = 2 1 1 2 4 s 5 s A = a 1 u 3 6 s s s (8) Compute det A. Is A invertible?
(b) Suppose that 4: :l. hi: 2] x=l:l B=liil 45. Find the inverse (if it exists) of A. In Problems 4548, let Write AX = B as a system of linear equations. 46. Find the inverse (if it exists) of B. (c) Show that if
47. Show that (A4)“1 = A. 3
48. Show that (A8)" = 3144. 3 = g
49. Find the inverse (if it exists) of
then
C [2 4] AX = B
“ 3 6 has inﬁnitely many solutions. Graph the two straight lines
. . . . associated with the corresponding system of linear equations, and
50' ﬁnd the inverse (If n extsts) 0f explain why the system has inﬁnitely many solutions.
1 0 0 (d) Find a column vector
13 = 0 1 0 b
0 O l = ‘
8 lb: l
51. Suppose that so that
_1 0 _2 AX = B
A = d D =
[ 7 ‘3 :I an [ “5 :I has no solutions. ;
Find X such that AX = D by 58' SUPPOSC that ‘
(u) solving the associated system oflinear equations and A _ a 8 X —. r bi
(b) using the inverse ofA. “ 2 4 ' ‘ and B = 1,2 (a) Show that when a 9e 4, AX = B has exactly one solution. (b) Suppose a = 4. Find conditions on b1 and b2 such that AX =
B has (i) inﬁnitely many solutions and (ii) no solutions (c) Explain your results in (a) and (b) graphically. In Problems 5 9—62, use the determinant to ﬁnd the in verse of A. 2 1 12
59.A=[3 _1] «LA—[0 3] 14 ~21
61.A=[51] 62.A_[ 3 2] 63. Use the determinant to determine whether A=lé a is invertible. If it is invertible, compute its inverse. In either case,
solve AX = 0. 64. Use the determinant to determine whether 1 1 ‘
B  l 2 1 l
is invertible. If it is invertible, compute its inverse. In either case,
solve BX = 0. 65. Use the determinant to determine whether 1 3
c = [1 3 ]
is invertible. If it is invertible, compute its inverse. In either case,
solve CX = 0. 66. Use the determinant to determine whether —3 6
D = [4 8 l
is invertible. If it is invertible, compute its inverse. In either case,
solve DX = 0. 1 9.2.4 In Problems 6770, ﬁnd the inverse matrix to each given matrix if
the inverse matrix exists. 2 —1 —1 ——1 3 1
67. A = 2 1 1 68. A = 2 2 3
—1 l ~1 > —1 1 2
—1 0 —1 —1 0 2 69. A = 0 —2 0 70. A = —1 —2
1 1 2 0 2 —1 1 9.2.5 In Problems 71—74, suppose that breeding occurs once a year and
that a census is taken at the end of each breeding season. 71. Assume that a population is divided into three age classes and
that 20% of the females age 0 and 70% of the females age I survive
until the end of the next breeding season. Assume further that
females age 1 have an average of 3.2 female offspring and females
age 2 have an average of 1.7 female offspring. If, at time 0, the
population consists of 2000 females age 0, 800 females age 1, and 200 females age 2. ﬁnd the Leslie matrix and the age distribution
at time 2. h 9.2 I Matrices 467 72. Assume that a population is divided into three age classes and
that 80% of the females age 0 and 10% of the females age I survive
until the end of the next breeding season. Assume further that
females age 1 have an average of 1.6 female offspring and females
age 2 have an average of 3.9 female offspring. If, at time 0, the
population consists of 1000 females age 0, 100 females age 1, and 20 females age 2. ﬁnd the Leslie matrix and the age distribution at
time 3. 73. Assume that a population is divided into four age classes and
that 70% of the females age 0,50% of the females age 1, and 10%
of the females age 2 survive until the end of the next breeding
season. Assume further that females age 2 have an average of 4.6
female offspring and females age 3 have an average of 3.7 female
offspring. If, at time 0, the population consists of 1500 females age
0, 500 females age 1, 250 females age 2, and 50 females age 3, ﬁnd
the Leslie matrix and the age distribution at time 2. 74. Assume that a population is divided into four age classes
and that 65% of the females age 0, 40% of the females age 1,
and 30% of the females age 2 survive until the end of the next
breeding season. Assume further that females age 1 have an
average of 2.8 female offspring, females age 2 have an average
of 7.2 female offspring, and females age 3 have an average of
3.7 female offspring. If, at time 0, the population consists of 1500
females age 0, 500 females age 1, 250 females age 2, and 50 females
age 3, ﬁnd the Leslie matrix and the age distribution at time 3. In Problems 75—76, assume the given Leslie matrix L. Determine the number of age classes in the population, the fraction of one yearolds that survive until the end of the next breeding season, and the average number of female offspring of a twoyearold female.
2 3 2 1 ‘ 0 5 0
0.4 0 O 0
75. L = O 0.6 0 0 76. L = 0.8 0 0 000.80 0030 In Problems 77~78, assume the given Leslie matrix L. Determine the number of age classes in the population. What fraction of
twoyearolds survive until the end of the next breeding season?
Determine the average number of female offspring of a oneyear
old female. 1 2.5 3 1.5 o 4.2 3.7
77.L= 0'9 0 0 0 78.L= 0.7 0 o
0 0.3 0 0 0 010
0 0 0.20 ' 79. Assume that the Leslie matrix is 1.2 3.2
L‘[0.8 0 ] Suppose that, at time: = 0, MW) = 100 and mm) = 0. Find the population vectors fort = 0, 1, 2, . . . , 10. Compute the successive
ratios
N00) N10)
t = and t) =
”U” N00 — 0 q“ M: — 0 for t = 1, 2, . . ., 10. What value do (100) and q, (t) approach as t —> 00? (Take a guess.) Compute the fraction of females age 0 . fort = O, 1, . . . , 10. Can you ﬁnd a stable age distribution? ML.
M
‘W— 468 Chapter 9 1 Linear Algebra and Analytic Geometry 80. Assume that the Leslie matrix is ratios
N 0) N10)
02 3 z = _.___“ and n = ——_—
L = [0.33 0] ‘1‘“) ~00 ~1) 4“ Nit: — 1) Suppose that at time! = o, No(0) = 10 and N,(0) = 5. Find the 10” = 1' 2'   .. 10 DO ‘10“) and ‘11“) converge? C°mP”‘° the . . .  fraction of females age 0 fort = 0. 1, . . . . 10. Describe the long
— ..., .C t the uc.e.srve ‘
population vectors for! 0, 1, 2. 10 ompu e s t s ter behavror of qo(r). ratios 82. Assume that the Leslie matrix is
N00) N10)
:1 ————— d t = ————
40(1) N004) an M) N104) L=[2 ii] for t = 1, 2. . . . . 10. What value do qo(t) and q1(t) approach as , _ _ _ ‘
I ——> 00? (Take a guess.) Compute the fraction of females: age 0 SUPPOSC that, at time t " 0' No(0) " 1 and N1 (0) " 1 Fmd the __ . . . n. population vectors fort = 0. 1, 2, . . . . 10. Compute the successive
for; __ 0, 1, . . . , 10. Can you ﬁnd a stable age distribution: ratios
81. Assume that the Leslie matrix is N . N ( )
0“) 1 f
I = ——— d t = —————————
L‘ 0 2 400 NW”) an (M) N,(:—1)
_ 0.6 0 fort = 1, 2. . . . . 10. Do (10(1) and ql(r) converge? Compute the
Suppose that. at time t = 0. mm) = 5 and mm) = 1. Find the fraction of females age 0 fort = 0, 1. . . . . 10. Describe the long population vectors for! = 0, 1. 2, . . . . 10. Compute the successive term behavior of (Mt). II 9.3 Linear Maps, Eigenvectors, and Eigenvalues In this section, we will denote vectors by boldface lowercase letters. Consider a map
of the form x ——> Ax (9.18) where A is a 2 x 2 matrix and x is a 2 x 1 column vector (or, simply, vector). Since Ax
is a 2 x 1 vector, this map takes a 2 x 1 vector and maps it into a 2 x 1 vector. That enables us to apply A repeatedly: We can compute A(Ax) = Azx, which is again a
2 x 1 vector, and so on. We will ﬁrst look at vectors, then at maps Ax, and ﬁnally at
iterates of the map A (i.e., Azx. A3x. and so on). According to the properties of matrix multiplication, the map (9.18) satisﬁes the
following conditions: 1. A(x + y) = Ax + Ay, and
2. A(Ax) = 1(Ax), where A is a scalar. Because of these two properties, we say that the map x —~> Ax is linear.
We saw an example of such a map in the previous section: If A is a 2 x 2 Leslie matrix and x is a population vector at time 0, then Ax represents the population
vector at time 11. Linear maps are important in other contexts as well, and we will encounter them
in Chapters 10 and 11. Here, we restrict our discussion to 2 x 2 matrices but point out
that we can generalize the discussion that follows to arbitrary n x n matrices. (These
topics are covered in courses on matrix or linear algebra.) 1 9.3.1 Graphical Representation Vectors We begin with a graphical representation of vectors. We assume that x is a
2 x 1 matrix. We call x a column vector or simply a vector. Since a 2 x 1 matrix has just
two components, we can represent a vector in the plane. For instance, to represent the vector
3
x ._"" 4 in the .nxz plane, we draw an arrow from the origin (0, 0) to the point (3, 4). as
illustrated in Figure 9.11. We see from Figure 9.11 that a vector has a length and a MM 486 Chapter 9 El Linear Algebra and Analytic Geometry which is the same fraction of zeroyearolds at time 0. Furthermore, since M > O,
we can choose ul so that both entries are positive (a condition that is needed if the
entries represent population sizes). We summarize this demonstration as follows: If L is a 2 x 2 Leslie matrix with eigenvalues A1 and A2, then the eigenvector
corresponding to the larger eigenvalue is a stable age distribution. For the matrix _ 1.5 2
" 0.08 0 20
1
is an eigenvector corresponding to the larger eigenvalue; thus, it is a stable age distri bution. In Section 9.2.5, we claimed that [210$] was a stable age distribution. In both cases, the fraction of zeroyear—olds is the same, namely, 20/21 = 2000/2100. That
is, both vectors represent the same proportion of zeroyear—olds in the population. We can check that [210$] = 100[21°], which shows that both vectors are eigenvectors. (Recall that if u is an eigenvector, then any vector on, a 75 0, is also an eigenvector.)
When we list a. stable age distribution, we make sure that both entries are positive
since they represent numbers of individuals in each age class. If A1 > p.21, then the population vector N (t) will converge to a stable age
distribution as t —> 00, provided that N (0) 9E “2. This follows from writing N (0) as a linear combination of the two eigenvectors 111 and U2 and applying L’ to the
result; that is, the vector L‘N(0) = L’ (mm + azuz) = alh'lul + azk'zuz =ah‘ x1 aA’ x2
11[Y1]+22 y2 where u1 = [2] and u; = [’3]. Here, a1 ¢ 0, since N (0) are u;. The fraction of
zeroyearolds at time t is akin + 02)»ng —) X1
0111161 + azl'zxz + aikih + 02%” x1 + Y1 ash—>00. Section9.3: Problems; I 9.3.1 (b) Show by direct calculation that A().x) = MAX).
1. Let
In Problems 3—8, represent each given vector x = 2 in the xlxz
A = I: 2 1 :l x = x1 ] and y = y1 plane, and determine its length and the angle that it forms with the
3 4 Y X: ' Y: positive xlaxis (measured counterclockwise).
(a) Show by direct calculation that .40: + y) = Ax + Ay. 3. x = [g] 4. x = [ "3] 5. x = [(3)]
(b) Show by direct calculation that A(}.x) = MAX). Met 6.x=[:i] 7.x=[‘~/f] 8.x=[_\/%] A = [:11 :12 ] , x = [:1 ] , and y = [y‘ ] In Problems 9—12, vectors are given in their polar coordinate
21 22 2 y 2 representation (length r, and angle a measured counterclockwise
(a) Show by direct calculation that A(x + y) = Ax + Ay. from the positive x, axis). Find the representation of the vector ['2] in Cartesian coordinates. 9...
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