MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the system of equations.
1) x1 + x2 + x3 = 7
1) B
x1 - x2 + 2x3 = 7
5x1 + x2 + x3 = 11
A) (4, 1, 2)
B) (1, 2, 4)
C) (4, 2, 1)
D) (1, 4, 2)
Determine whether the system is consistent.
2) x1 + x2 + x3 = 7
x1 - x2 + 2x3 = 7
2) B
5x1 + x2 + x3 = 11
A) No
B) Yes
Determine whether the matrix is in echelon form, reduced echelon form, or neither.
1 6
5 -7
3) 0 1 -4 -1
0 0
1 3
A) Echelon form
4)
B) Neither
3) A
C) Reduced echelon form
1 4 5 -7
4 1 -4 7
0 4 1 4
4) C
A) Echelon form
Find the indicated vector.
5) Let u = -2 . Find -9u.
-3
A)
18
-27
6) Let u =
B) Reduced echelon form
C) Neither
5) D
B)
C)
-18
27
D)
18
27
-18
-27
-3 . Find 7u.
2
A)
6) B
B)
21
-14
C)
D)
21
14
-21
14
1
-21
-14
Compute the product or state that it is undefined.
6
7) [-6 2 5]
0
-3
A)
B)
[-51]
-36
0
-15
7) A
C)
D)
[174]
[-36 0 -15]
Write the system as a vector equation or matrix equation as indicated.
8) Write the following system as a vector equation involving a linear combination of vectors.
3x1 - 5x2 - x3 = 2
5x1 +
3x3 = 6
A) x1 3
5
+ x2 -5 + x3 1 =
1
3
C) x1
3
5
8) A
+ x2
0
-5
+ x3
3
x1
x1
B) 3 x2 - 5 x2
x3
x3
2
6
x1
- x2
x3
=
2
6
0
3
5
2
D) x51 + x2 0 = 6
-1
3
0
-1 = 2
6
Solve the problem.
9) 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
A)
B)
x1
x1
-3
x2 = x3 3
0
x3
-3
x2 = x3 3
1
x3
C)
D)
x1
3
x2 = x3 -3
1
x3
x1
-3
x2 = 3
1
x3
2
9) B
10) Find the general solution of the simple homogeneous ʺsystemʺ below, which consists of a single
linear equation. Give your answer as a linear combination of vectors. Let x2 and x3 be free
10) C
variables.
-2x1 - 14x2 + 8x3 = 0
A)
x1
x 2 = x2
x3
4
-7
+
x
3 1
0
0
1
(with x2 , x3 free)
B)
x1
x1
x1
x2 = -7 x2 - 4 x2
x3
x3
(with x2 , x3 free)
x3
C)
x1
7
-4
x 2 = x 2 1 + x3 0
0
1
x3
(with x2 , x3 free)
D)
x1
x2 = x2
x3
-7
4
1 + x3 0
0
1
(with x2 , x3 free)
11) 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 sectorʹs income match its expenditures.
Find the general solution as a vector, with pa free.
A)
B)
pe
pe
0.308 pa
pm = 0.716 pa
pa
pa
0.465 pa
pm = 0.593 pa
pa
pa
C)
D)
pe
pe
0.607 pa
0.356 pa
pm = 0.686 pa
pa
pa
pm = 0.481 pa
pa
pa
3
11) D
12) 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.
A) x1 = 600 - x4
B) x1 = 600 - x4
C) x1 = 500 + x4
D) x1 = 600 + x5
x2 = 400 - x4
x2 = 400 + x4
x2 = 400 - x4
x2 = 400 - x5
x3 = 300 + x4
x3 = 300 - x4
x3 = 300 - x4
x3 = 300 - x5
x4 is free
x4 is free
x4 is free
x4 = 300
x5 = 300
x5 = 300
x5 = 200
x5 is free
13) For what values of h are the given vectors linearly independent?
1
-4
-6 , 24
1
h
A) Vectors are linearly independent for h = -4
B) Vectors are linearly dependent for all h
C) Vectors are linearly independent for all h
D) Vectors are linearly independent for h ≠ -4
1
-3
2
14) Let v1 = -3 , v2 = 8 , v3 = -2 .
-6
8
5
12) B
13) D
14) B
Determine if the set {v1 , v2 , v3 } is linearly independent.
A) Yes
B) No
Describe geometrically the effect of the transformation T.
0 0 0
15) Let A = 0 1 0 .
15) C
0 0 1
Define a transformation T by T(x) = Ax.
A) Projection onto the x2 -axis
B) Horizontal shear
C) Projection onto the x2 x 3 -plane
D) Vertical shear
4
Solve the problem.
16) Let T: ℛ2 -> ℛ2 be a linear transformation that maps u =
6 .
-8
Use the fact that T is linear to find the image of 3u + v.
A)
B)
C)
-5
-7
18
-2
-3
4
into -13
6
and maps v =
4
6
into
16) D
D)
-21
-6
-33
10
Determine whether the linear transformation T is one -to-one and whether it maps as specified.
17) T(x1 , x2 , x3 ) = (-2x2 - 2x3 , -2x1 + 8x2 + 4x3 , -x1 - 2x3 , 4x2 + 4x3 )
17) C
Determine whether the linear transformation T is one -to-one and whether it maps ℛ3 onto ℛ4 .
A) One-to-one; onto ℛ4
B) One-to-one; not onto ℛ4
C) Not one-to-one; not onto ℛ4
D) Not one-to-one; onto ℛ4
18) Let T be the linear transformation whose standard matrix is
1 -2 3
A = -1 3 -4 .
2 -2 -9
18)
Determine whether the linear transformation T is one -to-one and whether it maps ℛ3 onto ℛ3 .
A) Not one-to-one; not onto ℛ3
B) One-to-one; not onto ℛ3
C) Not one-to-one; onto ℛ3
D) One-to-one; onto ℛ3
Find the matrix product AB, if it is defined.
19) A = 0 -2 , B = -1 3 2 .
2 3
0 -3 1
19) B
A)
B)
0 -2 6
-3 -2 7
C) AB is undefined.
0 6 -2
-2 -3 7
D)
0 -6 -12
0 -9
3
The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, if the products are
defined.
20) A is 2 × 1, B is 1 × 1.
20) B
A) AB is 1 × 2, BA is 1 × 1.
B) AB is 2 × 1, BA is undefined.
C) AB is undefined, BA is 1× 2.
D) AB is 2 × 2, BA is 1 × 1.
Solve the system by using the inverse of the coefficient matrix.
21) 10x1 - 4x2 = -6
6x1 - x2 = 2
A) (-4, -1)
B) (4, 1)
21) C
C) (1, 4)
D) (-1, -4)
22) 2x1 - 6x2 = -6
3x1 + 2x2 = 13
A) (3, 2)
22) A
B) (2, 3)
C) (-3, -2)
5
D) (-2, -3)
Determine whether the matrix is invertible.
5 5 -5
23)
6 2 -6
-2 0 2
A) No
24)
23) A
B) Yes
6 7
1 18
A) Yes
24) A______
B) No
6