Exam
Name___________________________________

MULTIPLE CHOICE.  Choose the one alternative that best completes the statement or answers the question. 

Solve the system of equations. 

1) 

    -   + 3 = -8
2             +  = 0
   + 5  +   = 40 

1) 

_______ 

A) 

(8, 8, 0) 

B) 

(0, -8, -8) 

C) 

(-8, 0, 0) 

D) 

(0, 8, 0) 

2) 

 + 3 + 2 = 11
         4 + 9 = -12
                       = -4 

2) 

_______ 

A) 

(-4, 6, 1) 

B) 

(-4, 1, 6) 

C) 

(1, 6, -4) 

D) 

(1, -4, 6) 

3) 

  -  + 8 = -107
6	+  = 17
	3 - 5 = 89 

3) 

_______ 

A) 

(-5, 8, 13) 

B) 

(-5, -8, 13) 

C) 

(5, 8, -13) 

D) 

(5, -8, -13) 

4) 

4 - 	+ 3 = 12
2 	+ 9 = -5
	 + 4 + 6 = -32 

4) 

_______ 

A) 

(2, -7, -1) 

B) 

(2, -7, 1) 

C) 

(2, 7, -1) 

D) 

(2, 7, 1) 

5) 

 +   +   = 6
           -  = -2
         + 3 = 11 

5) 

_______ 

A) 

No solution 

B) 

(0, 1, 2) 

C) 

(1, 2, 3) 

D) 

(-1, 2, -3) 

6) 

  +  +   = 7
  -  + 2 = 7
5 +  +   = 11 

6) 

_______ 

A) 

(1, 4, 2) 

B) 

(4, 1, 2) 

C) 

(4, 2, 1) 

D) 

(1, 2, 4) 

7) 

 -  +  = 8
 +  +  = 6
 +  -  = -12 

7) 

_______ 

A) 

(-2, -1, 9) 

B) 

(2, -1, -9) 

C) 

(-2, -1, -9) 

D) 

(2, -1, 9) 

8) 

5 + 2 +  = -11
2 - 3 -  = 17
7 +  + 2 = -4 

8) 

_______ 

A) 

(3, 0, -4) 

B) 

(0, -6, 1) 

C) 

(0, 6, -1) 

D) 

(-3, 0, 4) 

9) 

7 + 7 +   = 1
  + 8 + 8 = 8
9 +   + 9 = 9 

9) 

_______ 

A) 

(0, 0, 1) 

B) 

(-1, 1, 1) 

C) 

(1, -1, 1) 

D) 

(0, 1, 0) 

10) 

2  +           = 0
   - 3 +  = 0
3  +   -  = 0 

10) 

______ 

A) 

(0, 1, 0) 

B) 

No solution 

C) 

(1, 0, 0) 

D) 

(0, 0, 0) 

Determine whether the system is consistent. 

11) 

  +  +   = 7
  -  + 2 = 7
5 +  +   = 11 

11) 

______ 

A) 

No 

B) 

Yes 

12) 

5 + 2 +  = -11
2 - 3 -  = 17
7 +  + 2 = -4 

12) 

______ 

A) 

Yes 

B) 

No 

13) 

4 -  + 3 = 12
2 	+ 9 = -5
 + 4 + 6 = -32 

13) 

______ 

A) 

Yes 

B) 

No 

14) 

2  +           = 0
   - 3 +  = 0
3  +   -  = 0 

14) 

______ 

A) 

No 

B) 

Yes 

15) 

 +   +   = 6
           -  = -2
         + 3 = 11 

15) 

______ 

A) 

No 

B) 

Yes 

16) 

      -    +  3 = -9
-5 + 5 - 15 = -2
     + 5     +  = -17 

16) 

______ 

A) 

No 

B) 

Yes 

17) 

   +   +   = 7
   -  + 2 = 7
2         + 3 = 15 

17) 

______ 

A) 

No 

B) 

Yes 

18) 

 + 3  +  2 = 11
        4  +  9 = -12
 + 7 + 11 = -11 

18) 

______ 

A) 

No 

B) 

Yes 

19) 

5 + 2 +  = -11
2 - 3 -  = 17
7  -            = 12 

19) 

______ 

A) 

No 

B) 

Yes 

20) 

          3            +   = -7
    +   + 2  -   = 12
 3             +  + 2 = 12
    +   + 5            = 26 

20) 

______ 

A) 

No 

B) 

Yes 

Determine whether the matrix is in echelon form, reduced echelon form, or neither. 

21) 

 

21) 

______ 

A) 

Neither 

B) 

Reduced echelon form 

C) 

Echelon form 

22) 

 

22) 

______ 

A) 

Reduced echelon form 

B) 

Neither 

C) 

Echelon form 

23) 

 

23) 

______ 

A) 

Echelon form 

B) 

Neither 

C) 

Reduced echelon form 

24) 

 

24) 

______ 

A) 

Reduced echelon form 

B) 

Echelon form 

C) 

Neither 

25) 

 

25) 

______ 

A) 

Reduced echelon form 

B) 

Neither 

C) 

Echelon form 

26) 

 

26) 

______ 

A) 

Neither 

B) 

Echelon form 

C) 

Reduced echelon form 

27) 

 

27) 

______ 

A) 

Reduced echelon form 

B) 

Neither 

C) 

Echelon form  

Use the row reduction algorithm to transform the matrix into echelon form or reduced echelon form as indicated. 

28) 

Find the echelon form of the given matrix.

28) 

______ 

A) 

 

B) 

 

C) 

 

D) 

 

29) 

Find the reduced echelon form of the given matrix.

29) 

______ 

A) 

 

B) 

 

C) 

 

D) 

 

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.  

30) 

 

30) 

______ 

A) 

 = -5 + 5
 = 7
 is free    

B) 

No solution 

C) 

 = -5 + 5
 is free   

D) 

(-5, 7) 

31) 

 

31) 

______ 

A) 

 = -25 + 11
 = 8 - 4
 = 6   

B) 

No solution 

C) 

 = -9 - 2 + 3
 is free
 is free  

D) 

 = -25 + 11
 = 8 - 4
 is free   

32) 

 

32) 

______ 

A) 

  = 5 -2 + 3
 = -6 - 4
 is free 

B) 

 = 5 - 2 + 3
 is free
 is free    

C) 

 = 17 + 11
 = -6 - 4
 = 0   

D) 

 = 17 + 11
 = -6 - 4
 is free  

33) 

 

33) 

______ 

A) 

 = 7 - 6
 = -2 + 2
 is free   

B) 

 = 7 - 6
 = -2 + 2
 = 0 

C) 

No solution 

D) 

 = 7 - 6
 is free
 = 1 +    

34) 

 

34) 

______ 

A) 

 = 11 - 4 - 3
 is free
 = 5 - 3
 = 0

B) 

 = - 4 +2 + 3  + 1
 is free
 = 5 - 3
 is free 

C) 

 = 11 - 4 - 3
 is free
 = 5 - 3
 is free

D) 

 = 11 - 4 - 3
 = 5 - 3
 is free 

35) 

 

35) 

______ 

A) 

 = -6 - 6 + 9
 is free
 is free
 =  -4
 =  -4 

B) 

No solution 

C) 

 = -6 - 6 + 9
 is free
 = -4
 =    - 1
 =  -4 

D) 

 = -6 - 6 +  - 2 + 5
 is free
 is free
 =    - 1
 =  -4

 

Find the indicated vector. 

36) 

Let u = , v = . Find u + v. 

36) 

______ 

A) 

 

B) 

 

C) 

 

D) 

 

37) 

Let u = , v = . Find u - v. 

37) 

______ 

A) 

 

B) 

 

C) 

 

D) 

 

38) 

Let u = , v = . Find v - u. 

38) 

______ 

A) 

 

B) 

 

C) 

 

D) 

 

39) 

Let u = . Find 6u. 

39) 

______ 

A) 

 

B) 

 

C) 

 

D) 

 

40) 

Let u = . Find 7u. 

40) 

______ 

A) 

 

B) 

 

C) 

 

D) 

 

41) 

Let u = . Find -9u. 

41) 

______ 

A) 

 

B) 

 

C) 

 

D) 

 

42) 

Let u = . Find -5u. 

42) 

______ 

A) 

 

B) 

 

C) 

 

D) 

 

43) 

Let u = , v = . Find 2u + v. 

43) 

______ 

A) 

 

B) 

 

C) 

 

D) 

 

Display the indicated vector(s) on an xy-graph. 

44) 

Let u =  and v = .  Display the vectors u, v, and u + v  on the same axes.

44) 

______ 

A) 

 

B) 

 

C) 

 

D) 

 

45) 

Let u =   Display the vector 2u  using the given axes.

45) 

______ 

A) 

 

B) 

 

C) 

 

D) 

 

Solve the problem. 

46) 

Let  = ,  = , and b =  .
Determine whether b can be written as a linear combination of  and . In other words, determine whether weights  and  exist, such that   +   = b. Determine the weights  and  if possible. 

46) 

______ 

A) 

No solution 

B) 

 = -3,  = 2 

C) 

 = -3,  = 3 

D) 

 = -2,  = 1 

47) 

Let  = ,  = ,  = , and b =  .
Determine whether b can be written as a linear combination of , , and . In other words, determine whether weights , , and  exist, such that   +   +   = b. Determine the weights , , and  if possible. 

47) 

______ 

A) 

 = 2,  = 1,  = -  

B) 

No solution 

C) 

 = -6,  = 0,  = 1 

D) 

 = -2,  = -1,  = 2 

SHORT ANSWER.  Write the word or phrase that best completes each statement or answers the question. 

48) 

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 =  and b = .

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. 

48)  

_____________ 

49) 

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 =  and b = .

Then a and b represent the "costs per dollar of income" for the two products.
Suppose the company manufactures  dollars worth of product A and  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  and , the dollars worth of each product produced. Include a vector equation as part of your solution. 

49)  

_____________ 

MULTIPLE CHOICE.  Choose the one alternative that best completes the statement or answers the question. 

Compute the product or state that it is undefined. 

50) 

[-6  2  5]   

50) 

______ 

A) 

[-36  0  -15] 

B) 

 

C) 

[-51] 

D) 

[174] 

51) 

51) 

______ 

A) 

 

B) 

[-63 -30] 

C) 

 

D) 

 

52) 

 	

52) 

______ 

A) 

Undefined 

B) 

 

C) 

 

D) 

 

53) 

   

53) 

______ 

A) 

 

B) 

 

C) 

 Undefined 

D) 

 

Write the system as a vector equation or matrix equation as indicated. 

54) 

Write the following system as a vector equation involving a linear combination of vectors.
	 3 - 5 -   = 2
	 5 +	 3 = 6 

54) 

______ 

A) 

 +  +  =  

B) 

 +  =  

C) 

3 - 5  -   =  

D) 

 +  +  =  

55) 

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. 
	 4  +  - 2  = 5
	 2 - 3	= 1 

55) 

______ 

A) 

  =  

B) 

  =  

C) 

  =  

D) 

   =  

Solve the problem. 

56) 

Let A =  and b =  .

Determine if the equation Ax = b is consistent for all possible , , . If the equation is not consistent for all possible , , , give a description of the set of all b for which the equation is consistent (i.e., a condition which must be satisfied by  

56) 

______ 

A) 

Equation is consistent for all possible , , . 

B) 

Equation is consistent for all , ,  satisfying 2 +   = 0. 

C) 

Equation is consistent for all , ,  satisfying -3 +  = 0. 

D) 

Equation is consistent for all , ,  satisfying 7 + 5 +  = 0. 

57) 

Let A =  and b =  .

Determine if the equation Ax = b is consistent for all possible , , . If the equation is not consistent for all possible , , , give a description of the set of all b for which the equation is consistent (i.e., a condition which must be satisfied by , ,  ). 

57) 

______ 

A) 

Equation is consistent for all , ,  satisfying -3 +  = 0. 

B) 

Equation is consistent for all , ,  satisfying - +  +  = 0. 

C) 

Equation is consistent for all possible , , . 

D) 

Equation is consistent for all , ,  satisfying 3 + 3 +  = 0. 

58) 

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  and  be free variables.
	 -2 - 14 + 8 = 0 

58) 

______ 

A) 

 = -7 - 4    (with ,  free) 

B) 

 =  +     (with ,  free)	 

C) 

 =  +     (with ,  free) 

D) 

 =  +     (with ,  free) 

59) 

Find the general solution of the homogeneous system below. Give your answer as a vector. 
	 + 2 - 3 = 0
	4 + 7 - 9 = 0
	- - 3 + 6 = 0 

59) 

______ 

A) 

 =    

B) 

 =   	  

C) 

 =   	  

D) 

 =    

60) 

Describe all solutions of Ax = b, where
	A =  and b = .

Describe the general solution in parametric vector form. 

60) 

______ 

A) 

 =  +   	   

B) 

 =  +  

C) 

 =  +   	   

D) 

 =  +   	   

61) 

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 , , and , respectively. If possible, find equilibrium prices that make each sector's income match its expenditures.
Find the general solution as a vector, with  free. 

61) 

______ 

A) 

 =  

B) 

 = 	 

C) 

 =  

D) 

 =  

62) 

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  be free.

62) 

______ 

A) 

 = 500 + 
 = 400 - 
 = 300 - 
 is free
 = 200  

B) 

 = 600 - 
 = 400 - 
 = 300 + 
 is free
 = 300  

C) 

 = 600 - 
 = 400 + 
 = 300 - 
 is free
 = 300  

D) 

 = 600 + 
 = 400 - 
 = 300 - 
 = 300
 is free  

63) 

Let  = ,  = ,  = .
	
Determine if the set {, , } is linearly independent. 

63) 

______ 

A) 

No 

B) 

Yes 

64) 

Determine if the columns of the matrix A =  are  linearly independent. 

64) 

______ 

A) 

Yes 

B) 

No 

65) 

For what values of h are the given vectors linearly independent?
	,  

65) 

______ 

A) 

Vectors are linearly independent for h = -4	 

B) 

Vectors are linearly independent for h ≠ -4 

C) 

Vectors are linearly independent for all h 

D) 

Vectors are linearly dependent for all h 

66) 

For what values of h are the given vectors linearly dependent?
	, , ,  

66) 

______ 

A) 

Vectors are linearly independent for all h 

B) 

Vectors are linearly dependent for h ≠ -24 

C) 

Vectors are linearly dependent for h = -24	 

D) 

Vectors are linearly dependent for all h  

67) 

Let A =  and u = .
Define a transformation T  ->  by T(x) = Ax. Find T(u), the image of u under the transformation T. 

67) 

______ 

A) 

 

B) 

 

C) 

  

D) 

 

68) 

Let T  ->  be a linear transformation that maps u =  into  and maps v =  into  . 
Use the fact that T is linear to find the image of  

68) 

______ 

A) 

 

B) 

 

C) 

 

D) 

 

69) 

Let A =  and b = .
Define a transformation T  ->  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. 

69) 

______ 

A) 

 

B) 

 

C) 

b is not in the range of the transformation T. 

D) 

 

70) 

Let A =  and b = .
Define a transformation T  ->  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. 

70) 

______ 

A) 

b is not in the range of the transformation T. 

B) 

 

C) 

 

D) 

 

Describe geometrically the effect of the transformation T.  

71) 

Let A = .
Define a transformation T by T(x) = Ax. 

71) 

______ 

A) 

Vertical shear 

B) 

Horizontal shear 

C) 

Projection onto -axis 

D) 

Projection onto -axis 

72) 

Let A = .
Define a transformation T by T(x) = Ax.  

72) 

______ 

A) 

Horizontal shear 

B) 

Projection onto the -plane 

C) 

Vertical shear 

D) 

Projection onto the -axis 

Solve the problem. 

73) 

The columns of  =  are  =  ,  =  ,  =  .

Suppose that T is a linear transformation from  into  such that

	T( ) =  , T( ) =  , and T( ) = .

Find a formula for the image of an arbitrary x =  in . 

73) 

______ 

A) 

T =  

B) 

T =  

C) 

T =  

D) 

T =  

Find the standard matrix of the linear transformation T.  

74) 

T  ->  rotates points (about the origin) through  π radians (with counterclockwise rotation for a positive angle). 

74) 

______ 

A) 

 

B) 

 

C) 

 

D) 

 

75) 

T  ->  first performs a vertical shear that maps  into  + 5, but leaves the vector  unchanged, then reflects the result through the horizontal -axis. 

75) 

______ 

A) 

 

B) 

 

C) 

 

D) 

 

Determine whether the linear transformation T is one-to-one and whether it maps as specified. 

76) 

Let T be the linear transformation whose standard matrix is
	A = .
Determine whether the linear transformation T is one-to-one and whether it maps  onto . 

76) 

______ 

A) 

Not one-to-one; onto  

B) 

One-to-one; onto  

C) 

Not one-to-one; not onto  

D) 

One-to-one; not onto  

77) 

T(, , ) = (- 2 - 2, -2 + 8 + 4, - - 2, 4 + 4)
Determine whether the linear transformation T is one-to-one and whether it maps  onto . 

77) 

______ 

A) 

One-to-one; onto  

B) 

Not one-to-one; onto  

C) 

Not one-to-one; not onto  

D) 

One-to-one; not onto  

Solve the problem. 

78) 

The table shows the amount (in g) of protein, carbohydrate, and fat supplied by one unit (100 g) of three different foods.  

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. 

78) 

______ 

A) 

0.302 units of Food 1,  0.238 units of Food 2,  0.085 units of Food 3 

B) 

0.280 units of Food 1,  0.192 units of Food 2,  0.164 units of Food 3 

C) 

0.360 units of Food 1,  0.204 units of Food 2,  0.055 units of Food 3 

D) 

0.326 units of Food 1,  0.247 units of Food 2,  0.059 units of Food 3 

79) 

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. 

79) 

______ 

A) 

City	 541,500
Suburbs 958,500 

B) 

City	 588,000
Suburbs 912,000 

C) 

City	 541,500
Suburbs 864,360 

D) 

City	 576,840
Suburbs 923,160 

 

1) 

D 

2) 

C 

3) 

C 

4) 

A 

5) 

C 

6) 

D 

7) 

A 

8) 

B 

9) 

A 

10) 

D 

11) 

B 

12) 

A 

13) 

A 

14) 

B 

15) 

B 

16) 

A 

17) 

A 

18) 

A 

19) 

A 

20) 

B 

21) 

C 

22) 

B 

23) 

B 

24) 

C 

25) 

C 

26) 

C 

27) 

C 

28) 

D 

29) 

C 

30) 

B 

31) 

B 

32) 

D 

33) 

A 

34) 

C 

35) 

A 

36) 

A 

37) 

B 

38) 

B 

39) 

D 

40) 

C 

41) 

D 

42) 

B 

43) 

D 

44) 

A 

45) 

D 

46) 

B 

47) 

B 

48) 

300a + 200b =  

300a + 200b lists the various costs for producing $300 worth of product A and $200 worth of product B, namely $200 for materials, $140 for labor, and $75 for overhead. 

49) 

a + b  = 
or

  +    = 

 = 300,  = 400 

50) 

C 

51) 

A 

52) 

D 

53) 

C 

54) 

D 

55) 

B 

56) 

A 

57) 

D 

58) 

C 

59) 

C 

60) 

A 

61) 

C 

62) 

C 

63) 

B 

64) 

B 

65) 

B 

66) 

D 

67) 

B 

68) 

C 

69) 

D 

70) 

A 

71) 

A 

72) 

B 

73) 

A 

74) 

B 

75) 

A 

76) 

B 

77) 

C 

78) 

D 

79) 

D