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Exam
Name___________________________________
1)
A = 4 0 1
2 -1-3
5 3 7 , B =-2 0 8 5
1 6 2 2
4 -1 0 3
1)
A)
-4-1 32 23
-17 -3 14 -1
21 11 46 52
B)
-4-1 32 23
-17 -3 14 -1
21 11 46 52
C)
-8 0 32 20
-5-6 14 8
-7 18 46 31
D)
-4-1 0 3
-12 -3 0 -9
28 -7 0 21
Find the matrix product AB, if it is defined.
2)
A =3-2
3 0 , B =0-2
3 6 .
2)
A)
0 4
9 0
B)
-6 0
27 -6
C)
-18 -6
-6 0
D)
-6-18
0-6
Find an LU factorization of the matrix A.
3)
A =2 4 5
411 5
4-124
3)
A)
A = 1 0 0
4 1 0
4-1 1
2 4 5
0 11 5
0 0 24
B)
A = 1 0 0
2 1 0
2 -3 1
3 4 5
0 -3 5
0 0 1
C)
A = 1 0 0
4 1 0
4-1 1
2 4 5
0 3 -5
0 0 -1
D)
A = 1 0 0
2 1 0
2 -3 1
2 4 5
0 3 -5
0 0 -1
Find the matrix product AB for the partitioned matrices.
4)
A =0 I
I F , B =W X
Y Z
4)
A)
0 Z
FY FZ
B)
Y Z
W + YF X + ZF
C)
X W + XF
Z Y + ZF
D)
Y Z
W + FY X + FZ
Decide whether or not the matrices are inverses of each other.
5)
-2 4
4-4 and
1
21
4
1
21
4
5)
A)
Yes
B)
No
Solve the system by using the inverse of the coefficient matrix.
6)
7x1- 2x2= 2
28x1- 8x2= 3
6)
A)
(4, 4)
B)
2
7-7
2x2, x2
C)
(2, 3)
D)
No solution
Find the matrix product AB, if it is defined.
7)
A =0-3
3 3 , B =-2 0
-1 1 .
7)
A)
-3 3
-3-9
B)
0 6
-3 3
C)
-6-6
3 6
D)
3-3
-9 3
Find the 3 × 3 matrix that produces the described transformation, using homogeneous coordinates.
8)
Reflect through the x-axis
8)
A)
-1 0 0
0 1 0
0 0 1
B)
1 0 0
0 1 0
0 0 1
C)
1 0 0
0-1 0
0 0 1
D)
-1 0 0
0 -1 0
0 0 1
Decide whether or not the matrices are inverses of each other.
9)
-5 1
-7 1 and
1
2-1
2
7
2-5
2
9)
A)
No
B)
Yes
Perform the matrix operation.
10)
Let C =2
-2
12 . Find (1/2) C.
10)
A)
1
-2
12
B)
4
-4
24
C)
1
-1
6
D)
2
-1
12
Find the inverse of the matrix A, if it exists.
11)
A =1 3 2
1 3 3
2 7 8
11)
A)
A-1=-1-3-2
-1-3-3
-2-7-8
B)
A-1=-3 10 -3
2-4 1
-1 1 0
C)
A-1=
11
31
2
11
31
3
1
21
71
8
D)
A-1 does not exist.
Find the matrix product AB, if it is defined.
12)
A =1 3 -2
2 0 3 , B =3 0
-2 1
0 3 .
12)
A)
-3-3
9 6
B)
AB is undefined.
C)
-3-3
6 9
D)
3-6 0
0 0 9
Decide whether or not the matrices are inverses of each other.
13)
10 1
-1 0 and 0 1
-110
13)
A)
No
B)
Yes
Find the inverse of the matrix, if it exists.
14)
A =-1-6
6 3
14)
A)
1
11 2
11
-2
11 -1
33
B)
1
11 -2
11
2
11 -1
33
C)
-1
33 2
11
-2
11 1
11
D)
-2
11 -1
33
1
11 2
11
Determine the production vector x that will satisfy demand in an economy with the given consumption matrix C and final
demand vector d. Round production levels to the nearest whole number.
15)
C =.4 .3
.1 .6 , d=52
76
15)
A)
x=44
51
B)
x=208
242
C)
x=3
25
D)
x=40
3
Find the inverse of the matrix A, if it exists.
16)
A =6-4 1
11 -7 2
5-3 1
16)
A)
A-1=
1
61
11 1
1
11 -1
71
2
1
5-1
31
B)
A-1=
1
11 2
11 - 4
5
11 8
73
6
5-1
31
C)
A-1=611 5
-4-7-3
1 2 1
D)
A-1 does not exist.
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.
17)
A is 4×4, B is 4×4.
17)
A)
AB is 1 × 1, BA is 1 × 1.
B)
AB is 4×4, BA is 4×4.
C)
AB is 4×8, BA is 4×8.
D)
AB is 8×4, BA is 8×4.
Decide whether or not the matrices are inverses of each other.
18)
9-2
7-2 and
0.5 0.5
-7
4-9
4
18)
A)
Yes
B)
No
Identify the indicated submatrix.
19)
A =
6 4 1
-6 0 -1
0 7-4
7 4 7
. Find A21.
19)
A)
-6
B)
1
-1
-4
C)
7 4
D)
4
Find the inverse of the matrix, if it exists.
20)
1 0 0
-1 1 0
1 1 1
20)
A)
1 1 1
0 1 1
0 0 1
B)
1-1 1
0 1 -1
001
C)
1 0 0
1 1 0
-2-1 1
D)
-1 0 0
-1-1 0
-1-1-1
Find the matrix product AB, if it is defined.
21)
A =3-2 1
0 4 -2 , B =4 0
-2 3 .
21)
A)
12 0
012
B)
AB is undefined.
C)
12 -6
-816
4-8
D)
12 -8 4
-616 -8
Perform the matrix operation.
22)
Let C =1
-3
2 and D =-1
3
-2 . Find C - 4D.
22)
A)
5
-6
4
B)
-3
9
-6
C)
5
-15
10
D)
-5
15
-10
Solve the system by using the inverse of the coefficient matrix.
23)
6x1+ 4x2= 4
3x1= -6
23)
A)
(-2, 4)
B)
No solution
C)
(4, -2)
D)
(-2, -4)
Perform the matrix operation.
24)
Let A =1 3
2 6 and B =0 4
-1 6 . Find 4A + B.
24)
A)
4 7
712
B)
416
730
C)
428
448
D)
416
112
Find the transpose of the matrix.
25)
9 8 9 8
0 -7 0 -7
25)
A)
0 -7 0 -7
9 8 9 8
B)
9 0
8-7
9 0
8-7
C)
8 9 8 9
-7 0 -7 0
D)
0 9
-7 8
0 9
-7 8
8
Perform the matrix operation.
26)
Let A =-3 1
0 2 . Find 2A.
26)
A)
-1 3
2 4
B)
-6 2
0 4
C)
-6 2
0 2
D)
-6 1
0 2
Identify the indicated submatrix.
27)
A =0 1 -7-5
7-102
2 5 -2 0 . Find A12.
27)
A)
7
B)
1
C)
2 5 -2
D)
-5
2
Perform the matrix operation.
28)
Let A =-2 7
-2-8 and B =2 6
-2 9 . Find A - B.
28)
A)
0 1
0-17
B)
0-1
-417
C)
-4 1
0-17
D)
4 1
-4 1
Determine the production vector x that will satisfy demand in an economy with the given consumption matrix C and final
demand vector d. Round production levels to the nearest whole number.
29)
C =.2 .1 .1
.3 .2 .3
.4 .1 .3 , d=213
322
296
29)
A)
x=104
217
206
B)
x=109
105
90
C)
x=481
892
825
D)
x=728
988
-312
Determine the rank of the matrix.
30)
1 -2 2 -4
2 -4 7 -4
-3 6 -612
30)
A)
4
B)
1
C)
3
D)
2
Find the inverse of the matrix, if it exists.
31)
A =-2-5
1 0
31)
A)
-2
51
-1
50
B)
0 1
-1
5-2
5
C)
-1
5-2
5
0 1
D)
0- 1
1
5-2
5
Find the 4 × 4 matrix that produces the described transformation, using homogeneous coordinates.
32)
Translation by the vector (4, -7, -9)
32)
A)
1 0 0 4
0 1 0 -7
0 0 1 -9
0 0 0 1
B)
1 0 0 -4
0 1 0 7
0 0 1 9
0 0 0 1
C)
0 0 0 4
0 0 0 -7
0 0 0 -9
0 0 0 1
D)
4 0 0 0
0-7 0 0
0 0 -9 0
0 0 0 1
10
Find an LU factorization of the matrix A.
33)
A =3-1
-18 9
33)
A)
A = 1 0
-6 1 3 1
0 -3
B)
A = 1 0
3 1 -6-1
0 3
C)
A = 1 0
-6 1 3-1
0 3
D)
A = 1 0
6 1 -3-1
0 -3
Determine whether the matrix is invertible.
34)
6 7
118
34)
A)
No
B)
Yes
Solve the system by using the inverse of the coefficient matrix.
35)
2x1+ 6x2=2
2x1-x2= -5
35)
A)
(-1, 2)
B)
(-2, 1)
C)
(2, -1)
D)
(1, -2)
Find the matrix product AB, if it is defined.
36)
A =0-2
2 3 , B =-1 3 2
0-3 1 .
36)
A)
0-2 6
-3-2 7
B)
0-6-12
0-9 3
C)
AB is undefined.
D)
0 6 -2
-2-3 7
11
Find a basis for the null space of the matrix.
37)
A =
1 0 -5 0 -2
0 1 3 0 3
0 0 0 1 1
0 0 0 0 0
37)
A)
1
0
-5
0
-2
,
0
1
3
0
3
B)
1
0
0
0
,
0
1
0
0
,
0
0
1
0
C)
-5
3
1
0
0
,
-2
3
0
-1
1
D)
5
-3
1
0
0
,
2
-3
0
-1
1
The vector x is in a subspace H with a basis = {b1, b2}. Find the -coordinate vector of x.
38)
b1= 1
-2, b2=-5
3 , x=17
-13
38)
A)
2
-3
B)
-3
2
C)
-4
1
D)
-2
3
Find the inverse of the matrix A, if it exists.
39)
A =5-1 5
5 0 4
10 -1 9
39)
A)
A-1=5 5 10
-1 0 -1
5 4 9
B)
A-1 does not exist.
C)
A-1=
1 0 4
5
0 1 -1
04
50
D)
A-1=1 0 4
5
0 1 -1
0 0 0
Find the transpose of the matrix.
40)
4 6
-6 0
-5 5
40)
A)
4-6-5
6 0 5
B)
-5 5
-6 0
4 6
C)
6 0 5
4-6-5
D)
6 4
0 -6
5-5
Decide whether or not the matrices are inverses of each other.
41)
2-1 0
-1 1 -2
1 0 -1 and 1-1 2
-3-2 4
-1 1 1
41)
A)
Yes
B)
No
Find the 3 × 3 matrix that produces the described composite 2D transformation, using homogeneous coordinates.
42)
Rotate points through 45° and then scale the x-coordinate by 0.2 and the y-coordinate by 0.4.
42)
A)
0.1 20.1 2 0
-0.2 20.2 2 0
0 0 1
B)
0 -0.2 0
0.4 0 0
0 0 1
C)
0.1 2-0.1 2 0
0.2 20.2 2 0
0 0 1
D)
0.1 -0.2 2 0
0.1 20.2 0
0 0 1
Find the matrix product AB, if it is defined.
43)
A =1 0
0 2 , B =1 2 -2
2-2 2 .
43)
A)
4-4 4
1 2 -2
B)
AB is undefined.
C)
1 2 -2
4-4 4
D)
1 0 0
0-4 4
Find the inverse of the matrix, if it exists.
44)
A =4-2
2-1
44)
A)
1
3-2
3
1
6-1
3
B)
A is not invertible
C)
1
6-1
3
1
3-2
3
D)
1
61
3
-1
3-2
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.
45)
A is 2×1, B is 1×1.
45)
A)
AB is 1×2, BA is 1×1.
B)
AB is 2×2, BA is 1×1.
C)
AB is undefined, BA is 1×2.
D)
AB is 2×1, BA is undefined.
Determine whether b is in the column space of A.
46)
A =-1 0 2
5 8 -10
-3-3 6 , b=-4
5
3
46)
A)
Yes
B)
No
Solve the equation Ax = b by using the LU factorization given for A.
47)
A = 3 -1 2
-6 4 -5
9 5 6 , b= 6
-3
2
A = 1 0 0
-2 1 0
3 4 1
3-1 2
0 2 -1
0 0 4
47)
A)
x= 22
-7
15
B)
x= 49
-38
32
C)
x= 25
-58
51
D)
x= 10
-2
-13
The vector x is in a subspace H with a basis = {b1, b2}. Find the -coordinate vector of x.
48)
b1= 2
-2
4 , b2= 6
1
-3, x=26
9
-23
48)
A)
5
-2
B)
-2
5
C)
-2
5
0
D)
2
-5
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.
49)
A is 2×3, B is 3×2.
49)
A)
AB is 3×3, BA is 2×2.
B)
AB is 2×2, BA is undefined.
C)
AB is 2×2, BA is 3×3.
D)
AB is undefined, BA is 3×3.
Decide whether or not the matrices are inverses of each other.
50)
6-5
-3 5 and
1
31
3
1
52
5
50)
A)
No
B)
Yes
Solve the system by using the inverse of the coefficient matrix.
51)
2x1- 4x2= -2
3x1+ 4x2= -23
51)
A)
(5, 2)
B)
(2, 5)
C)
(-2, 5)
D)
(-5, -2)
Decide whether or not the matrices are inverses of each other.
52)
5 3
3 2 and 2-3
-3 5
52)
A)
Yes
B)
No
53)
9 4
4 4 and -0.2 0.2
0.2 -0.45
53)
A)
No
B)
Yes
Solve the system by using the inverse of the coefficient matrix.
54)
-5x1+ 3x2= 8
2x1- 4x2= -20
54)
A)
(-2, -6)
B)
(2, 6)
C)
(6, 2)
D)
(-6, -2)
Determine whether b is in the column space of A.
55)
A = 1 2 -3
1 4 -6
-3-2 5 , b=3
2
-5
55)
A)
No
B)
Yes
Find the 3 × 3 matrix that produces the described composite 2D transformation, using homogeneous coordinates.
56)
Translate by (9, 7), and then reflect through the line y = x.
56)
A)
0 1 9
1 0 7
0 0 1
B)
-1 0 -9
0 -1-7
0 0 1
C)
0 7 1
9 0 0
0 0 1
D)
0 1 7
1 0 9
0 0 1
Solve the system by using the inverse of the coefficient matrix.
57)
2x1- 6x2= -6
3x1+ 2x2= 13
57)
A)
(2, 3)
B)
(-3, -2)
C)
(3, 2)
D)
(-2, -3)
Find the 4 × 4 matrix that produces the described transformation, using homogeneous coordinates.
58)
Rotation about the y-axis through an angle of 60°
58)
A)
0.5 0 3/2 0
0 1 0 0
-3/2 0 0.5 0
0 0 0 1
B)
1 0 0 0
0 0.5 3/2 0
0-3/2 0.5 0
0 0 0 1
C)
0.5 3/2 0 0
-3/2 0.5 0 0
0 0 1 0
0 0 0 1
D)
3/2 0 0.5 0
0 1 0 0
-0.5 0 3/2 0
0 0 0 1
Solve the problem.
59)
Compute the matrix of the transformation that performs the shear transformation x
Ax for
A =10.17
0 1 and then scales all y-coordinates by a factor of 0.63.
59)
A)
20.17
01.63
B)
10.1071
00.63
C)
0.63 0.1071
0 1
D)
10.17
00.63
Find the inverse of the matrix A, if it exists.
60)
A =1 0 8
1 2 3
2 5 3
60)
A)
A-1=1 1 2
0 2 5
8 3 3
B)
A-1=-1 0 -8
-1-2-3
-2-5-3
C)
A-1 does not exist.
D)
A-1=9-40 16
-3 13 -5
-1 5 -2
Solve the problem.
61)
Compute the matrix of the transformation that performs the shear transformation x
Ax for
A = 1 0.21
0 1 and then scales all x-coordinates by a factor of 0.68.
61)
A)
0.68 0.21
0 1
B)
1 0.21
0 0.68
C)
1.68 0.21
0 2
D)
0.68 0.1428
0 1
Determine the rank of the matrix.
62)
10-3 0 4
01-3 0 2
00 0 1 1
00 0 0 0
62)
A)
4
B)
2
C)
5
D)
3
Find the inverse of the matrix, if it exists.
63)
A =-3 1
0 6
63)
A)
-1
31
18
01
6
B)
1
61
18
0-1
3
C)
-1
3-1
18
01
6
D)
01
6
-1
31
18
19
64)
A =0 5
6 3
64)
A)
-1
10 -1
6
-1
50
B)
01
6
1
5-1
10
C)
1
50
-1
10 1
6
D)
-1
10 1
6
1
50
Solve the equation Ax = b by using the LU factorization given for A.
65)
A =
1 2 4 3
-1-3-1-4
2 1 19 3
1 5 -9 7
, b=
2
0
4
3
A =
1 0 0 0
-1 1 0 0
2 3 1 0
1 -3-2 1
1 2 4 3
0-1 3 -1
0 0 2 0
0 0 0 1
65)
A)
x=
2
-2
8
-3
B)
x=
27
9
8
-3
C)
x=
41
-6
-3
-5
D)
x=
27
-18
89
-13
Solve the system by using the inverse of the coefficient matrix.
66)
5x1+ 3x2= 3
2x1+ 5x2= 24
66)
A)
(-3, 6)
B)
(-3, -6)
C)
No solution
D)
(6, -3)
20
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.
67)
A is 2×1, B is 2×1.
67)
A)
AB is 2×2, BA is 1×1.
B)
AB is undefined, BA is undefined.
C)
AB is 1×2, BA is 2×1.
D)
AB is 2×1, BA is 1×2.
Perform the matrix operation.
68)
Let A =-10 3
7-9 and B =0 0
0 0 . Find A + B.
68)
A)
-10 3
7-9
B)
10 -3
-7 9
C)
Undefined
D)
0 0
0 0
Find a basis for the null space of the matrix.
69)
A =1 0 -5-2
0 1 7 -4
0 0 0 0
69)
A)
1
0
-5
-2
,
0
1
7
-4
B)
-5
7
1
0
,
-2
-4
0
1
C)
5
-7
1
0
,
2
4
0
1
D)
1
0
0, 0
1
0
Find the inverse of the matrix, if it exists.
70)
A =-4 0
-3-3
70)
A)
-1
40
1
4-1
3
B)
A is not invertible
C)
-1
30
1
4-1
4
D)
-1
40
-1
4-1
3
Perform the matrix operation.
71)
Let A =-1 2 and B =1 0 . Find 3A + 4B.
71)
A)
-3 4
B)
-1 4
C)
1 6
D)
2 2
22
Find the inverse of the matrix A, if it exists.
72)
A =022
-204
070
72)
A)
A-1=
1-1
2-2
7
0 0 1
7
1
20-1
7
B)
A-1=
- 1 -1
2-2
7
-1
701
7
1
20 0
C)
A-1 does not exist.
D)
A-1=
1 0 1
2
-1
20 0
-2
71
7-1
7
Find the matrix product AB, if it is defined.
73)
A =-1 3
2 2 , B =-2 0
-1 4 .
73)
A)
12 -1
8-6
B)
-112
-6 8
C)
2-6
-1 5
D)
2 0
-2 8
Perform the matrix operation.
74)
Let B =-1 4 7 -3. Find -4B.
74)
A)
4-16 -28 12
B)
-416 28 -12
C)
4 4 7 -3
D)
-3 2 5 -5
Solve the system by using the inverse of the coefficient matrix.
75)
10x1- 4x2= -6
6x1-x2= 2
75)
A)
(1, 4)
B)
(-4, -1)
C)
(4, 1)
D)
(-1, -4)
Find the matrix product AB, if it is defined.
76)
A =-1 3
1 6 , B =0-2 7
1-3 2 .
76)
A)
AB is undefined.
B)
0-6
21 1
-18 12
C)
3 6 -7
-20 -119
D)
3-7-1
6-20 19
Find a basis for the column space of the matrix.
77)
B =
1 0 -5 0 -5
0 1 5 0 4
0 0 0 1 1
0 0 0 0 0
77)
A)
1
0
0
0
,
0
1
0
0
B)
5
-5
1
0
0
,
5
-4
0
-1
1
C)
1
0
0
0
,
0
1
0
0
,
-5
5
0
0
D)
1
0
0
0
,
0
1
0
0
,
0
0
1
0
78)
B = 1 -2 2 -3
2 -4 9 -2
-3 6 -6 9
78)
A)
1
2
-3, -2
-4
6
B)
2
1
0
0
,
23
5
0
-4
5
1
C)
1
0
0, 0
1
0
D)
1
2
-3, 2
9
-6
Determine whether the matrix is invertible.
79)
5 5 -5
6 2 -6
-2 0 2
79)
A)
No
B)
Yes
Decide whether or not the matrices are inverses of each other.
80)
-5-1
6 0 and
01
6
-15
6
80)
A)
No
B)
Yes
Find the inverse of the matrix A, if it exists.
81)
A =1 1 1
2 1 1
2 2 3
81)
A)
A-1=-1-1-1
-2-1-1
-2-2-3
B)
A-1=-1 1 0
4-1-1
-2 0 1
C)
A-1=
1 1 1
1
21 1
1
21
21
3
D)
A-1 does not exist.
Perform the matrix operation.
82)
Let A =-1 7
-5 1
4 9 and B =5 6
-3-9
-5 7 . Find A + B.
82)
A)
413
-8-8
-116
B)
4 1
-8-8
-116
C)
413
8 1
-1-16
D)
-6 1
-2-2
9 0
Find the 3 × 3 matrix that produces the described transformation, using homogeneous coordinates.
83)
(x, y)
(x +5, y +4)
83)
A)
5 0 0
0 4 0
0 0 1
B)
1 0 4
0 1 5
0 0 1
C)
1 0 5
0 1 4
0 0 0
D)
1 0 5
0 1 4
0 0 1
26
Answer Key
Testname: C2
Answer Key
Testname: C2
