Exam
Name___________________________________
1)
A company manufactures two products. For $1.00 worth of product A, the company
spends $0.40 on materials, $0.30 on labor, and $0.15 on overhead. For $1.00 worth of
product B, the company spends $0.40 on materials, $0.25 on labor, and $0.15 on overhead.
Let
a=0.40
0.30
0.15 and b =0.40
0.25
0.15 .
Then a and b represent the “costs per dollar of income” for the two products.
Evaluate 300a+200b and give an economic interpretation of the result.
1)
2)
A company manufactures two products. For $1.00 worth of product A, the company
spends $0.45 on materials, $0.20 on labor, and $0.10 on overhead. For $1.00 worth of
product B, the company spends $0.45 on materials, $0.20 on labor, and $0.15 on overhead.
Let
a=0.45
0.20
0.10 and b =0.45
0.20
0.15 .
Then a and b represent the “costs per dollar of income” for the two products.
Suppose the company manufactures x1 dollars worth of product A and x2 dollars worth of
product B and that its total costs for materials are $315, its total costs for labor are $140, and
its total costs for overhead are $90.
Determine x1 and x2, the dollars worth of each product produced. Include a vector
equation as part of your solution.
2)
1
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
3)
Let A = 1 –3 2
–2 5 –1
3 –6–3 and b=b1
b2
b3
.
Determine if the equation Ax = b is consistent for all possible b1, b2, b3. If the equation is not
consistent for all possible b1, b2, b3, give a description of the set of all b for which the equation is
consistent (i.e., a condition which must be satisfied by b1, b2, b3 ).
A)
Equation is consistent for all possible b1, b2, b3.
B)
Equation is consistent for all b1, b2, b3 satisfying 3b1+3b2+b3= 0.
C)
Equation is consistent for all b1, b2, b3 satisfying –3b1+b3= 0.
D)
Equation is consistent for all b1, b2, b3 satisfying –b1+b2+b3= 0.
4)
The columns of I3=1 0 0
0 1 0
0 0 1 are e1=1
0
0 , e2=0
1
0 , e3=0
0
1 .
Suppose that T is a linear transformation from
3 into
2 such that
T( e1) =6
–3 , T( e2) =2
0 , and T( e3) =–2
1 .
Find a formula for the image of an arbitrary x =x1
x2
x3
in
3.
A)
Tx1
x2
x3
=6x1+2x2–2x3
–3x1+x3
B)
Tx1
x2
x3
=6x1+2x2–2x3
2x1
–3x1+x3
C)
Tx1
x2
x3
=6x1–3x2
2x1
D)
Tx1
x2
x3
=6x1–3x2
2x1
2x2+x3
Determine whether the system is consistent.
5)
4x1–x2+ 3x3= 12
2x1+ 9x3= –5
x1+ 4x2+ 6x3= –32
A)
No
B)
Yes
6)
Find the general solution of the simple homogeneous “system” below, which consists of a single
linear equation. Give your answer as a linear combination of vectors. Let x2 and x3 be free
variables.
–2x1–14x2+8x3= 0
A)
x1
x2
x3
= –7x1
x2
x3
– 4 x1
x2
x3
(with x2, x3 free)
B)
x1
x2
x3
=x2–7
1
0 +x34
0
1 (with x2, x3 free)
C)
x1
x2
x3
=x2–7
0
1 +x34
1
0 (with x2, x3 free)
D)
x1
x2
x3
=x27
1
0 +x3–4
0
1 (with x2, x3 free)
7)
x1+x2+x3= 6
x1–x3= –2
x2+ 3x3= 11
A)
(1, 2, 3)
B)
(–1, 2, –3)
C)
(0, 1, 2)
D)
No solution
3
8)
4x1–x2+ 3x3= 12
2x1+ 9x3= –5
x1+ 4x2+ 6x3= –32
A)
(2, 7, –1)
B)
(2, 7, 1)
C)
(2, –7, –1)
D)
(2, –7, 1)
Determine whether the system is consistent.
9)
x1+ 3x2+ 2x3= 11
4x2+ 9x3= –12
x1+ 7x2+ 11x3= –11
A)
Yes
B)
No
Use the row reduction algorithm to transform the matrix into echelon form or reduced echelon form as indicated.
10)
Find the reduced echelon form of the given matrix.
1 4 –5 1 2
2 5 –4–1 4
–3–9 9 2 2
10)
A)
1 4 –5 0 –2
0 1 –2 0 –4
0 0 0 1 4
B)
1 0 0 0 14
0 1 0 0 –4
0 0 0 1 4
C)
1 0 3 0 14
0 1 –2 0 –4
0 0 0 1 4
D)
1 4 –5 1 2
0 1 –2 1 0
0 0 0 1 4
4
Compute the product or state that it is undefined.
11)
[–6 2 5] 6
0
–3
11)
A)
[174]
B)
–36
0
–15
C)
[–36 0 –15]
D)
[–51]
Use the row reduction algorithm to transform the matrix into echelon form or reduced echelon form as indicated.
12)
Find the echelon form of the given matrix.
1 4 –2 3
–3–11 9 –5
–2 3 4 –3
12)
A)
1 4 –2 3
0 1 3 4
0 0 –33 –41
B)
1 4 –2 3
0 1 3 4
0 0 –33 0
C)
1 4 –2 3
0 1 3 4
0 0 –11 –8
D)
1 4 –2 3
0 1 3 4
011 0 3
Display the indicated vector(s) on an xy–graph.
13)
Let u=–3
5 and v=–2
2 . Display the vectors u, v, and u + v on the same axes.
13)
5
A)
B)
C)
D)
6
The augmented matrix is given for a system of equations. If the system is consistent, find the general solution. Otherwise
state that there is no solution.
14)
1 6 6 –1 2 5
0 0 0 –4 3 4
0 0 0 0 –2 8
14)
A)
No solution
B)
x1= –6x2–6x3+9
x2 is free
x3 is free
x4= –4
x5= –4
C)
x1= –6x2–6x3+9
x2 is free
x3= –4
x4=3
4x5– 1
x5= –4
D)
x1= –6x2–6x3+x4– 2x5+5
x2 is free
x3 is free
x4=3
4x5– 1
x5= –4
7
15)
The network in the figure shows the traffic flow (in vehicles per hour) over several one–way streets
in the downtown area of a certain city during a typical lunch time. Determine the general flow
pattern for the network.
In other words, find the general solution of the system of equations that describes the flow. In your
general solution let x4 be free.
15)
A)
x1= 500 +x4
x2= 400 –x4
x3= 300 –x4
x4 is free
x5= 200
B)
x1= 600 –x4
x2= 400 –x4
x3= 300 +x4
x4 is free
x5= 300
C)
x1= 600 +x5
x2= 400 –x5
x3= 300 –x5
x4= 300
x5 is free
D)
x1= 600 –x4
x2= 400 +x4
x3= 300 –x4
x4 is free
x5= 300
Find the standard matrix of the linear transformation T.
16)
T:
2->
2 rotates points (about the origin) through 7
4 radians (with counterclockwise rotation
for a positive angle).
16)
A)
3
33
3
–3
33
3
B)
1 1
–1 1
C)
2
22
2
–2
22
2
D)
–2
2–2
2
–2
22
2
Write the system as a vector equation or matrix equation as indicated.
17)
Write the following system as a matrix equation involving the product of a matrix and a vector on
the left side and a vector on the right side.
4x1+ x2–2x3=5
2x1–3x2=1
17)
A)
4 1 –2
2 3 1
x1
x2
x3
=5
1
B)
4 2
1 –3
–2 0
x1
x2=5
1
C)
x1x2x3
2–3 0
4
1
–2=5
1
D)
4 1 –2
2–3 0
x1
x2
x3
=5
1
Display the indicated vector(s) on an xy–graph.
18)
Let u= 5
–4 Display the vector 2u using the given axes.
18)
A)
B)
9
C)
D)
Solve the problem.
19)
For what values of h are the given vectors linearly independent?
1
–6
1, –4
24
h
19)
A)
Vectors are linearly dependent for all h
B)
Vectors are linearly independent for h –4
C)
Vectors are linearly independent for all h
D)
Vectors are linearly independent for h = –4
Answer:
B
Explanation:
A)
B)
Determine whether the system is consistent.
20)
x1+x2+x3= 7
x1–x2+ 2x3= 7
5x1+x2+x3= 11
20)
A)
Yes
B)
No
Answer:
A
Explanation:
A)
B)
10
Answer:
C
Explanation:
A)
B)
Solve the system of equations.
21)
x1+ 3x2+ 2x3= 11
4x2+ 9x3= –12
x3= –4
21)
A)
(–4, 1, 6)
B)
(1, –4, 6)
C)
(–4, 6, 1)
D)
(1, 6, –4)
Determine whether the system is consistent.
22)
5x1+ 2x2+x3= –11
2x1– 3x2–x3= 17
7x1+x2+ 2x3= –4
22)
A)
No
B)
Yes
The augmented matrix is given for a system of equations. If the system is consistent, find the general solution. Otherwise
state that there is no solution.
23)
1 2 –3 5
0 1 4 –6
0 0 0 0
23)
A)
x1=17 + 11x3
x2= –6– 4x3
x3= 0
B)
x1=5–2x2+ 3x3
x2= –6– 4x3
x3 is free
C)
x1=5– 2x2+ 3x3
x2 is free
x3 is free
D)
x1=17 + 11x3
x2= –6– 4x3
x3 is free
11
Solve the problem.
24)
For what values of h are the given vectors linearly dependent?
–1
4
6 , 5
2
–3, 6
2
6, –24
–8
h
24)
A)
Vectors are linearly dependent for h –24
B)
Vectors are linearly independent for all h
C)
Vectors are linearly dependent for h = –24
D)
Vectors are linearly dependent for all h
Compute the product or state that it is undefined.
25)
8–5
2 8
–6 6
–1
1
25)
A)
–8 2
5 8
–6 6
B)
–13
6
12
C)
Undefined
D)
–8–5
–2 8
6 6
Determine whether the system is consistent.
26)
x1+x2+x3= 6
x1–x3= –2
x2+ 3x3= 11
26)
A)
No
B)
Yes
Describe geometrically the effect of the transformation T.
27)
Let A = 1 0
6 1 .
Define a transformation T by T(x) = Ax.
27)
A)
Horizontal shear
B)
Projection onto x2–axis
C)
Vertical shear
D)
Projection onto x1–axis
Solve the problem.
28)
Let a1= 1
2
–3, a2=–3
–4
1 , a3=2
1
6, and b=–4
2
2 .
Determine whether b can be written as a linear combination of a1, a2, and a3. In other words,
determine whether weights x1, x2, and x3 exist, such that x1a1+x2a2+x3a3=b. Determine the
weights x1, x2, and x3 if possible.
28)
A)
x1= –2, x2= –1, x3=2
B)
x1= 2, x2= 1, x3= – 3
2
C)
x1= –6, x2= 0, x3= 1
D)
No solution
29)
8–2
5–2
4–5
–4
–6
4
29)
A)
–32 8
–30 12
16 –20
B)
–46 0
C)
–24
–18
–4
D)
Undefined
Determine whether the matrix is in echelon form, reduced echelon form, or neither.
30)
1 0 –4–2
0 1 4 5
0 0 0 0
0 0 0 0
30)
A)
Echelon form
B)
Reduced echelon form
C)
Neither
Solve the problem.
31)
Determine if the columns of the matrix A =–2 1 4
4 0 –4
2 7 12 are linearly independent.
31)
A)
Yes
B)
No
32)
Let A = 1 –3 2
–2 5 –1
3 –4–6 and b=b1
b2
b3
.
Determine if the equation Ax = b is consistent for all possible b1, b2, b3. If the equation is not
consistent for all possible b1, b2, b3, give a description of the set of all b for which the equation is
consistent (i.e., a condition which must be satisfied by b1, b2, b3).
32)
A)
Equation is consistent for all b1, b2, b3 satisfying 2b1+b2= 0.
B)
Equation is consistent for all possible b1, b2, b3.
C)
Equation is consistent for all b1, b2, b3 satisfying 7b1+ 5b2+b3= 0.
D)
Equation is consistent for all b1, b2, b3 satisfying –3b1+b3= 0.
Solve the system of equations.
33)
7x1+ 7x2+x3= 1
x1+ 8x2+ 8x3= 8
9x1+x2+ 9x3= 9
33)
A)
(–1, 1, 1)
B)
(0, 0, 1)
C)
(0, 1, 0)
D)
(1, –1, 1)
Determine whether the system is consistent.
34)
x1–x2+3x3= –9
–5x1+ 5x2– 15x3= –2
x1+ 5x2+x3= –17
34)
A)
Yes
B)
No
14
Compute the product or state that it is undefined.
35)
–8 1 –1
3 8 1
6
–7
8
35)
A)
–8 1 –1
3 8 1
6–7 8
B)
–30
–63
C)
[–63 –30]
D)
–63
–30
Determine whether the matrix is in echelon form, reduced echelon form, or neither.
36)
1 4 5 –7
0 1 –4–5
0 3 1 4
36)
A)
Neither
B)
Echelon form
C)
Reduced echelon form
Solve the problem.
37)
Describe all solutions of Ax=b, where
A = 2 –5 3
–2 6 –5
–4 7 0 and b=–1
6
–13 .
Describe the general solution in parametric vector form.
37)
A)
x1
x2
x3
=7/2
2
1 +x312
5
0
B)
x1
x2
x3
=12
5
0 +x37/2
2
0
C)
x1
x2
x3
=–1
5
0 +x3–1
2
1
D)
x1
x2
x3
=12
5
0 +x37/2
2
1
The augmented matrix is given for a system of equations. If the system is consistent, find the general solution. Otherwise
state that there is no solution.
38)
1 –5–5
0 0 7
38)
A)
x1= –5+ 5x2
x2=7
x3 is free
B)
(–5, 7)
C)
x1= –5+ 5x2
x2 is free
D)
No solution
Determine whether the system is consistent.
39)
5x1+ 2x2+x3= –11
2x1– 3x2–x3= 17
7x1–x2= 12
39)
A)
Yes
B)
No
Solve the problem.
40)
Let A = 1 –3 0
–4 0 2
1–5 4 and b=18
–10
32 .
Define a transformation T:
3->
3 by T(x) = Ax.
If possible, find a vector x whose image under T is b. Otherwise, state that b is not in the range of
the transformation T.
40)
A)
3
–5
1
B)
b is not in the range of the transformation T.
C)
3
–5
0
D)
3
1
–5
Describe geometrically the effect of the transformation T.
41)
Let A =0 0 0
0 1 0
0 0 1 .
Define a transformation T by T(x) = Ax.
41)
A)
Horizontal shear
B)
Vertical shear
C)
Projection onto the x2–axis
D)
Projection onto the x2x3–plane
Determine whether the matrix is in echelon form, reduced echelon form, or neither.
42)
1–5–5 3
0 0 2–2
0 0 0 5
0 0 0 0
42)
A)
Neither
B)
Reduced echelon form
C)
Echelon form
Find the indicated vector.
43)
Let u=2
4, v=2
9. Find u–v.
43)
A)
–7
2
B)
–2
–7
C)
4
13
D)
0
–5
Solve the problem.
44)
Suppose an economy consists of three sectors: Energy (E), Manufacturing (M), and Agriculture (A).
Sector E sells 70% of its output to M and 30% to A.
Sector M sells 30% of its output to E, 50% to A, and retains the rest.
Sector A sells 15% of its output to E, 30% to M, and retains the rest.
Denote the prices (dollar values) of the total annual outputs of the Energy, Manufacturing, and
Agriculture sectors by pe, pm, and pa, respectively. If possible, find equilibrium prices that make
each sector’s income match its expenditures.
Find the general solution as a vector, with pa free.
44)
A)
pe
pm
pa
=0.465 pa
0.593 pa
pa
B)
pe
pm
pa
=0.607 pa
0.481 pa
pa
C)
pe
pm
pa
=0.308 pa
0.716 pa
pa
D)
pe
pm
pa
=0.356 pa
0.686 pa
pa
Find the indicated vector.
45)
Let u=–2
–3. Find –9u.
45)
A)
–18
27
B)
–18
–27
C)
18
27
D)
18
–27
Solve the problem.
46)
Let T:
2->
2 be a linear transformation that maps u=–3
4 into –13
6 and maps v=4
6 into
6
–8 .
Use the fact that T is linear to find the image of 3u+v.
46)
A)
–21
–6
B)
–7
–2
C)
–5
18
D)
–33
10
47)
Let v1= 1
–3
8,v2=–3
8
5, v3= 2
–2
–6.
Determine if the set {v1, v2, v3} is linearly independent.
47)
A)
No
B)
Yes
Find the indicated vector.
48)
Let u=–3
2. Find 7u.
48)
A)
21
14
B)
–21
–14
C)
21
–14
D)
–21
14
Determine whether the matrix is in echelon form, reduced echelon form, or neither.
49)
1 0 0 –7
7 1 0 5
0 4 1 7
49)
A)
Echelon form
B)
Reduced echelon form
C)
Neither
50)
1 4 5 –7
4 1 –4 7
0 4 1 4
50)
A)
Neither
B)
Echelon form
C)
Reduced echelon form
The augmented matrix is given for a system of equations. If the system is consistent, find the general solution. Otherwise
state that there is no solution.
51)
1 4 –2–3 1
0 0 1 3 5
–1–4–2–9–21
51)
A)
x1=11 – 4x2–3x3
x2=5–3x3
x3 is free
B)
x1=11 – 4x2–3x4
x2 is free
x3=5–3x4
x4= 0
C)
x1=11 – 4x2–3x4
x2 is free
x3=5–3x4
x4 is free
D)
x1= – 4x2+2x3+ 3 x4+ 1
x2 is free
x3=5–3x4
x4 is free
Solve the system of equations.
52)
x1–x2+x3= 8
x1+x2+x3= 6
x1+x2–x3= –12
52)
A)
(2, –1, –9)
B)
(2, –1, 9)
C)
(–2, –1, –9)
D)
(–2, –1, 9)
Determine whether the system is consistent.
53)
2x1+x2= 0
x1– 3x2+x3= 0
3x1+x2–x3= 0
53)
A)
Yes
B)
No
20
Find the indicated vector.
54)
Let u=5
–6, v=–2
–3. Find 2u+v.
54)
A)
–2
–5
B)
6
–9
C)
12
–9
D)
8
–15
Solve the problem.
55)
The population of a city in 2000 was 600,000 while the population of the suburbs of that city in 2000
was 900,000.
Suppose that demographic studies show that each year about 5% of the city’s population moves to
the suburbs (and 95% stays in the city), while 2% of the suburban population moves to the city (and
98% remains in the suburbs).
Compute the population of the city and of the suburbs in the year 2002. For simplicity, ignore other
influences on the population such as births, deaths, and migration into and out of the city/suburban
region.
55)
A)
City: 541,500
Suburbs: 864,360
B)
City: 588,000
Suburbs: 912,000
C)
City: 576,840
Suburbs: 923,160
D)
City: 541,500
Suburbs: 958,500
Determine whether the matrix is in echelon form, reduced echelon form, or neither.
56)
1 3 7 –7
0 1 –4–3
0 0 0 0
56)
A)
Neither
B)
Echelon form
C)
Reduced echelon form
The augmented matrix is given for a system of equations. If the system is consistent, find the general solution. Otherwise
state that there is no solution.
57)
1 2 –3–9
0 1 4 8
0 0 0 6
57)
A)
x1= –25 + 11x3
x2=8– 4x3
x3 is free
B)
x1= –25 + 11x3
x2=8– 4x3
x3=6
C)
No solution
D)
x1= –9– 2x2+ 3x3
x2 is free
x3 is free
Write the system as a vector equation or matrix equation as indicated.
58)
Write the following system as a vector equation involving a linear combination of vectors.
3x1–5x2–x3=2
5x1+3x3=6
58)
A)
x13
5
–1+x25
0
3=2
6
0
B)
x13
5+x2–5
0 +x3–1
3=2
6
C)
3x1
x2
x3
–5x1
x2
x3
–x1
x2
x3
=2
6
0
D)
x13
5+x2–5
1 +x3 1
3=2
6
Solve the problem.
59)
Let A =–1 8 –1
3 4 2 and u=8
2
–3.
Define a transformation T:
3->
2 by T(x) = Ax. Find T(u), the image of u under the
transformation T.
59)
A)
11
26
B)
16
24
–3
C)
–816 3
24 8–6
D)
48
18
22
Determine whether the system is consistent.
60)
3x2+x4= –7
x1+x2+2x3–x4=12
3x1+x3+2x4=12
x1+x2+5x3=26
60)
A)
No
B)
Yes
Solve the system of equations.
61)
2x1+x2= 0
x1– 3x2+x3= 0
3x1+x2–x3= 0
61)
A)
(0, 0, 0)
B)
(0, 1, 0)
C)
(1, 0, 0)
D)
No solution
Determine whether the linear transformation T is one–to–one and whether it maps as specified.
62)
Let T be the linear transformation whose standard matrix is
A = 1 –2 3
–1 3 –4
2–2–9.
Determine whether the linear transformation T is one–to–one and whether it maps
3 onto
3.
62)
A)
One–to–one; onto
3
B)
One–to–one; not onto
3
C)
Not one–to–one; not onto
3
D)
Not one–to–one; onto
3
Solve the system of equations.
63)
x1–x2+ 3x3= –8
2x1+x3= 0
x1+ 5x2+x3= 40
63)
A)
(8, 8, 0)
B)
(0, 8, 0)
C)
(–8, 0, 0)
D)
(0, –8, –8)
23
64)
x1+x2+x3= 7
x1–x2+ 2x3= 7
5x1+x2+x3= 11
64)
A)
(4, 2, 1)
B)
(1, 2, 4)
C)
(1, 4, 2)
D)
(4, 1, 2)
The augmented matrix is given for a system of equations. If the system is consistent, find the general solution. Otherwise
state that there is no solution.
65)
1 0 6 7
0 1 –2–2
0 0 0 0
65)
A)
x1=7– 6x3
x2= –2+ 2x3
x3 is free
B)
No solution
C)
x1=7– 6x3
x2 is free
x3=1+1
2x2
D)
x1=7– 6x3
x2= –2+ 2x3
x3= 0
Solve the system of equations.
66)
5x1+ 2x2+x3= –11
2x1– 3x2–x3= 17
7x1+x2+ 2x3= –4
66)
A)
(0, 6, –1)
B)
(0, –6, 1)
C)
(3, 0, –4)
D)
(–3, 0, 4)
Find the standard matrix of the linear transformation T.
67)
T:
2->
2 first performs a vertical shear that maps e1 into e1+5e2, but leaves the vector e2
unchanged, then reflects the result through the horizontal x1–axis.
67)
A)
1 0
–5–1
B)
1 5
0 –1
C)
–1 0
5–1
D)
–1–5
0 1
24
Solve the problem.
68)
Let A = 1 –3 2
–3 4 –1
2 –5 3 and b=3
–5
–2.
Define a transformation T:
3->
3 by T(x) = Ax.
If possible, find a vector x whose image under T is b. Otherwise, state that b is not in the range of
the transformation T.
68)
A)
2
0
–2
B)
–5
–5
0
C)
b is not in the range of the transformation T.
D)
6
3
3
69)
Let a1=4
1
–2, a2=–1
3
3, and b=–14
3
12 .
Determine whether b can be written as a linear combination of a1 and a2. In other words,
determine whether weights x1 and x2 exist, such that x1a1+x2a2=b. Determine the weights x1
and x2 if possible.
69)
A)
x1= –3, x2=3
B)
x1= –3, x2=2
C)
No solution
D)
x1= –2, x2=1
Determine whether the system is consistent.
70)
x1+x2+x3= 7
x1–x2+ 2x3= 7
2x1+ 3x3= 15
70)
A)
Yes
B)
No
25
Find the indicated vector.
71)
Let u=8
8, v=4
–1. Find v–u.
71)
A)
–9
–4
B)
12
7
C)
–4
–9
D)
0
–5
72)
Let u=–8
7. Find –5u.
72)
A)
40
–35
B)
–40
35
C)
–40
–35
D)
40
35
73)
Let u=–5
–2. Find 6u.
73)
A)
–30
–12
B)
30
–12
C)
30
12
D)
–30
12
Solve the problem.
74)
The table shows the amount (in g) of protein, carbohydrate, and fat supplied by one unit (100 g) of
three different foods.
Food 1 Food 2 Food 3
Protein 15 35 25
Carbohydrate 45 30 50
Fat 6 4 1
Betty would like to prepare a meal using some combination of these three foods. She would like the
meal to contain 15 g of protein, 25 g of carbohydrate, and 3 g of fat. How many units of each food
should she use so that the meal will contain the desired amounts of protein, carbohydrate, and fat?
Round to 3 decimal places.
74)
A)
0.302 units of Food 1, 0.238 units of Food 2, 0.085 units of Food 3
B)
0.326 units of Food 1, 0.247 units of Food 2, 0.059 units of Food 3
C)
0.360 units of Food 1, 0.204 units of Food 2, 0.055 units of Food 3
D)
0.280 units of Food 1, 0.192 units of Food 2, 0.164 units of Food 3
Solve the system of equations.
75)
x1–x2+ 8x3= –107
6x1+x3= 17
3x2– 5x3= 89
75)
A)
(–5, 8, 13)
B)
(5, –8, –13)
C)
(–5, –8, 13)
D)
(5, 8, –13)
Determine whether the linear transformation T is one–to–one and whether it maps as specified.
76)
T(x1, x2, x3) = (–2x2–2x3, –2x1+8x2+4x3, –x1– 2x3, 4x2+4x3)
Determine whether the linear transformation T is one–to–one and whether it maps
3 onto
4.
76)
A)
One–to–one; not onto
4
B)
One–to–one; onto
4
C)
Not one–to–one; onto
4
D)
Not one–to–one; not onto
4
27
Determine whether the matrix is in echelon form, reduced echelon form, or neither.
77)
1 6 5 –7
0 1 –4–1
0 0 1 3
77)
A)
Echelon form
B)
Neither
C)
Reduced echelon form
Find the indicated vector.
78)
Let u=–4
–1, v=–7
–9. Find u+v.
78)
A)
3
8
B)
–11
–10
C)
–13
–8
D)
–5
–16
Solve the problem.
79)
Find the general solution of the homogeneous system below. Give your answer as a vector.
x1+ 2x2– 3x3= 0
4x1+ 7x2– 9x3= 0
–x1–3x2+6x3= 0
79)
A)
x1
x2
x3
=–3
3
1
B)
x1
x2
x3
=x3 3
–3
1
C)
x1
x2
x3
=x3–3
3
0
D)
x1
x2
x3
=x3–3
3
1
Answer Key
Testname: C1
Answer Key