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MATH1002 Linear Algebra

创建日期：2024-05-10 14:15:55

所属分类：数学mathematics

标签： MATH1002

Linear Algebra

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MATH1002
Linear Algebra
This examination has two sections: Multiple Choice and Extended Answer.
The Multiple Choice Section is worth 50% of the total examination;
there are 20 questions; the questions are of equal value;
all questions may be attempted.
Answers to the Multiple Choice questions must be entered on
the Multiple Choice Answer Sheet.
The Extended Answer Section is worth 50% of the total examination;
there are 3 questions; the questions are of equal value;
all questions may be attempted;
working must be shown.
University approved calculators may be used.
THE QUESTION PAPER MUST NOT BE REMOVED FROM THE
EXAMINATION ROOM.
Marker’s use
only
Page 1 of 22
CC1451A Semester 1, 2016 Page 2 of 22
Multiple Choice Section
In each question, choose at most one option.
Your answers must be entered on the Multiple Choice Answer Sheet.
1. The cosine of the angle between the vectors i+ 2j+ 2k and 3i+ 4k is equal to
(a)
14
225
(b)
11
15
(c)
11√
15
(d)
15
11
(e)
225
11
2. The length of the vector 3 (i− j− k) is equal to
(a) 9 (b)
√
18 (c)
√
27 (d) 18 (e)
√
3
3. If u = −i+ 2j+ 2k and v = 3i− 2j+ k then u · v is equal to
(a) 6i+ 7j− 4k (b) −5 (c) 6i− 7j+ 4k
(d) 9 (e) −9
4. The unit vector in the direction of 2i− j− 2k is
(a)
1
6
(2i− j− 2k) (b) i+ j+ k (c) i− j− k
(d)
1
5
(2i− j− 2k) (e) 1
3
(2i− j− 2k)
5. If a = −i+ j+ k and b = i− j+ k then a× b is equal to
(a) 0 (b) −2i− 2j (c) 0
(d) 2i+ 2j (e) i− j− 2k
CC1451A Semester 1, 2016 Page 3 of 22
This blank page may be used for rough working; it will not be marked. Be
sure to enter your answers on the Multiple Choice Answer Sheet.
CC1451A Semester 1, 2016 Page 4 of 22
6. The system of equations
2x − 2y − 2z = 4
x − 5z = 0
5x − 4y − 9z = 8
has the general solution
(a) x = 5, y = 2, z = 1.
(b) x = 0, y = −1, z = 0.
(c) x = 1 + 2t, y = 2 + 9t, z = t where t ∈ R.
(d) x = 5t, y = 4t + 2, z = t where t ∈ R.
(e) x = 5t, y = 4t− 2, z = t where t ∈ R.
7. Let A = P D1 P
−1, and B = P D2 P−1 where D1 =
[
1 0
0 2
]
, D2 =
[
3 0
0 1
]
, and
P =
[
1 1
1 0
]
. Then (AB)5 is
(a)
[
25 35 − 25
0 35
]
. (b)
[ −35 25 − 35
0 −25
]
. (c)
[
35 0
0 25
]
.
(d)
[
25 1
0 35
]
. (e)
[ −25 −35 + 25
0 −35
]
.
8. The determinant of the matrix
 2 2 03 −1 2
1 2 −1
 is equal to
(a) −4 (b) 16 (c) 4 (d) −16 (e) 0
9. Which one of the following matrices is in reduced row echelon form?
(a)
 1 0 00 0 0
0 0 1
 (b)
 1 −3 0 −10 6 1 4
0 0 0 2
 (c) [ 0 0 1 2
1 1 0 1
]
(d)
[
1 3 2
0 1 0
]
(e) None of the above.
CC1451A Semester 1, 2016 Page 5 of 22
This blank page may be used for rough working; it will not be marked. Be
sure to enter your answers on the Multiple Choice Answer Sheet.
CC1451A Semester 1, 2016 Page 6 of 22
10. Let B be a 4× 4 matrix and suppose that det(B) = 2 . Then det
(
− 1√
2
B
)
is
(a) 1/2 (b) −1/
√
2 (c) −1 (d) −
√
2 (e) −2
11. The line through the point (−1, 2, 3) perpendicular to the plane x + 2y + 7z = 0
is given by
(a) r = 4j+ 10k+ s(i+ 2j+ 7k), s ∈ R.
(b) r = −i+ 2j+ 3k+ s(i− 2j+ 7k), s ∈ R.
(c) x + 1 =
y − 2
−2 =
z − 3
7
.
(d) x− 1 = y − 2
2
=
z − 3
7
.
(e) None of the above.
12. Which one of the following implications may be false for some square matrices A, B?
(a) ABA = BAB =⇒ (AB)3 = (BA)3
(b) (A−B)2 = A2 − 2AB + B2 =⇒ AB = BA
(c) AB = BA =⇒ (A + B)(A−B) = A2 −B2
(d) det(A) = 0 =⇒ det(AB) = 0
(e) (AB)2 = 0 =⇒ AB = 0
13. If A, B and C are 3× 3 invertible matrices, and ABC =
 2 0 00 2 0
0 0 2
, then the inverse
of B is
(a) 2C−1A−1. (b)
1
8
AC. (c)
1
2
A−1C−1. (d)
1
2
C−1A−1. (e)
1
2
CA.
14. Which one of the following statements is false, given that A is a matrix of size 3× 3,
B is a matrix of size 3× 2, and C is a matrix of size 2× 3?
(a) A2 + BC is a 3× 3 matrix. (b) ACB is defined.
(c) 2A + CB is not defined. (d) (BC)2 is a 3× 3 matrix.
(e) B(A−BC) is not defined.
CC1451A Semester 1, 2016 Page 7 of 22
This blank page may be used for rough working; it will not be marked. Be
sure to enter your answers on the Multiple Choice Answer Sheet.
CC1451A Semester 1, 2016 Page 8 of 22
15. Consider the three planes with equations
P1 : x + 2y − z = 1,
P2 : −2x − 4y + 2z = 2,
P3 : −3x − 6y + 3z = 6.
Which of the following is true?
(a) None of the planes are parallel to each other.
(b) P1 and P2 are parallel to each other but not parallel to P3.
(c) P1 and P3 are parallel to each other but not parallel to P2.
(d) P2 and P3 are parallel to each other but not parallel to P1.
(e) All of the planes are parallel to each other.
16. Consider the following system of equations:
x + y + z − w = 0
−y + 2z − w = 0
2x + 6z − 4w = 0
Which one of the following statements about this system is true?
(a) There is a unique solution.
(b) The general solution is expressed using 1 parameter.
(c) The general solution is expressed using 2 parameters.
(d) The general solution is expressed using 3 or more parameters.
(e) There is no solution.
17. Which one of the following sequences of row operations, when applied to the matrix[
a b c
d e f
]
, produces the matrix
[
d− a e− b f − c
3a 3b 3c
]
?
(a) First R1 := R1 −R2, then R2 := 3R2, then R1 ↔ R2.
(b) First R1 ↔ R2 then R1 := 3R1, then R1 := R1 −R2.
(c) First R2 := R2 −R1, then R1 ↔ R2, then R1 := 3R1.
(d) First R1 := 3R1, then R1 ↔ R2, then R1 := R1 −R2.
(e) First R1 ↔ R2, then R1 := R1 −R2, then R2 := 3R2.
CC1451A Semester 1, 2016 Page 9 of 22
This blank page may be used for rough working; it will not be marked. Be
sure to enter your answers on the Multiple Choice Answer Sheet.
CC1451A Semester 1, 2016 Page 10 of 22
18. Suppose v and w are two non-zero vectors lying in this page:
v w
Which of the following is true?
(a) (v ×w) · v is a non-zero scalar.
(b) v and v ×w are parallel.
(c) (v ×w)× v is perpendicular to both v and w.
(d) (w × v)× (v ×w) is parallel to v but not w.
(e) v ×w points upwards, towards the ceiling.
19. The two lines given by the respective parametric scalar equations
x = 3 + t
y = −5 + 2t
z = −5 − t
 t ∈ R and x = −3 − 2sy = −2 − s
z = 1 + 2s
 s ∈ R
(a) do not intersect. (b) intersect at the point (7, 3,−9).
(c) intersect at the point (−2,−15, 0). (d) intersect at the point (−3,−2, 1).
(e) coincide.
20. Which one of the following statements about the matrix
[ −1 1
1 −1
]
is true?
(a) −2 is an eigenvalue with eigenspace
{[
t
−t
] ∣∣∣ t ∈ R}.
(b) 0 is an eigenvalue with eigenspace
{[
t
2t
] ∣∣∣ t ∈ R}.
(c) 0 is an eigenvalue with eigenspace
{[−t
t
] ∣∣∣ t ∈ R}.
(d) 1 is an eigenvalue with eigenspace
{[−t
−t
] ∣∣∣ t ∈ R}.
(e) 2 is an eigenvalue with eigenspace
{[−t
t
] ∣∣∣ t ∈ R}.
CC1451A Semester 1, 2016 Page 11 of 22
This blank page may be used for rough working; it will not be marked. Be
sure to enter your answers on the Multiple Choice Answer Sheet.
CC1451A Semester 1, 2016 Page 12 of 22
This blank page may be used for rough working; it will not be marked. Be
sure to enter your answers on the Multiple Choice Answer Sheet.
CC1451A Semester 1, 2016 Page 13 of 22
This blank page may be used for rough working; it will not be marked.
End of Multiple Choice Section
Make sure that your answers are entered on the Multiple Choice Answer Sheet
The Extended Answer Section begins on the next page
CC1451A Semester 1, 2016 Page 14 of 22
Extended Answer Section
There are three questions in this section, each with a number of parts. Write your answers
in the space provided. If you need more space there are extra pages at the end of the
examination paper.
1. (a) What is the Cartesian equation of the plane through the points P (1, 0, 0), Q(0, 2, 0)
and R(0, 0, 3)?
(b) Let L1 be the line given by the Cartesian equations
x− 1
2
=
y
−3 =
z − 2
1
.
Find the closest point R on the line L1 to the point P (1, 2, 1).
CC1451A Semester 1, 2016 Page 15 of 22
(c) Let L be the line given by the parametric vector equation
r = r0 + tv,
where r0 = i + 2j + 3k and v = −i + j − k, and let P be the plane given by the
vector equation
(r− r1) · n = 0,
where r1 = 3j+ 3k and n = i+ 3j+ k. Find the point where L and P intersect.
This is the end of Question 1.
CC1451A Semester 1, 2016 Page 16 of 22
2. (a) Find the value(s) of the parameter a for which the following system of linear equa-
tions has a unique solution.
x + ay + z = 2
x + y + az = 1
x + y + z = 1
(b) Let M =
 0 1 10 1 0
−1 −1 0
. Find M−1.
CC1451A Semester 1, 2016 Page 17 of 22
(c) Let P (x) denote a polynomial function of the form P (x) = Ax2 + Bx + C, where
A,B,C are real constants. Show that for any real numbers b1, b2, b3, there is a
unique choice of A,B,C such that the following equations all hold:
P (1) = b1,
P (2) = b2,
P (3) = b3.
This is the end of Question 2.
CC1451A Semester 1, 2016 Page 18 of 22
3. (a) Let M =
0 1 00 −1 0
0 2 1
. Find the eigenvalues of M .
(b) With M as in the previous part, find an invertible matrix P such that P−1MP is a
diagonal matrix. You do not need to find P−1.
CC1451A Semester 1, 2016 Page 19 of 22
(c) Suppose that A is a 3×3 matrix which has eigenvalues −t, 0 and t for some positive
real number t. Prove that A3 is a scalar multiple of A.
This is the end of Question 3.
CC1451A Semester 1, 2016 Page 20 of 22
There are no more questions.
More space is available on the next page.
CC1451A Semester 1, 2016 Page 21 of 22
This blank page may be used if you need more space for your answers.
CC1451A Semester 1, 2016 Page 22 of 22
This blank page may be used if you need more space for your answers.
End of Extended Answer Section
This is the last page of the question paper.

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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 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