Nanyang Technological University
SPMS/Division of Mathematical Sciences
NTU H3 Numbers and Matrices 2013
Tutorial 8 solution
1. Let A be an invertible matrix.
(i) Use mathematical induction to prove that An is invertible and (An )1 = (A1 )n
for n = 1, 2,
Nanyang Technological University
Semester II, 2007-2008
H3 Numbers and Matrices
Examination
INSTRUCTIONS TO CANDIDATES
This examination paper contains SIX (6) questions and comprises FIFTEEN (15 )
printed pages.
Answer ALL questions. The marks for each
Nanyang Technological University
Semester II, 2008-2009
H3 Numbers and Matrices
Examination
INSTRUCTIONS TO CANDIDATES
This examination paper contains SIX (6) questions and comprises FIFTEEN (15 )
printed pages.
Answer ALL questions. The marks for each
Nanyang Technological University
Semester II, 2009-2010
H3 Numbers and Matrices
Final exam
INSTRUCTIONS TO CANDIDATES
Time allowed: 120 minutes.
This test paper contains SIX (6) questions and comprises NINE (9) printed pages.
Answer all questions. The
Nanyang Technological University
Semester II, 2007-2008
H3 Numbers and Matrices
Examination
INSTRUCTIONS TO CANDIDATES
This examination paper contains SIX (6) questions and comprises FIFTEEN (15 )
printed pages.
Answer ALL questions. The marks for each
Nanyang Technological University
Semester II, 2010-2011
NTU H3 Numbers and Matrices
Examination
12th May 2011
3.30pm 5.30pm
Name:
Question
1
2
3
4
5
6
Total
Marks
INSTRUCTIONS TO CANDIDATES
This examination paper contains SIX (6) questions and comprises
Nanyang Technological University
Semester II, 20062007
H3 maths Numbers and Matrices
Mid term test 1
Date: 14 February 2007
Time allowed: 1 hour.
INSTRUCTIONS TO CANDIDATES
This examination paper contains FOUR (4) questions and comprises FIVE (5)
printed
Nanyang Technological University
Semester II, 2009-2010
H3 Numbers and Matrices
Final exam
INSTRUCTIONS TO CANDIDATES
Time allowed: 120 minutes.
This test paper contains SIX (6) questions and comprises NINE (9) printed pages.
Answer all questions. The
Nanyang Technological University
Semester II, 20062007
H3 Mathematics Numbers and Matrices
Final Examination
Date: 16 May 2007
Time allowed: 2 hours.
INSTRUCTIONS TO CANDIDATES
This examination paper contains SIX (6) questions and comprises TEN (10)
prin
1
Solutions to Final Exam of 2008
1. Question 1.
(a) (i) The symbol a|b means that there exists some integer c such that b = ac.
(ii) The symbol a b (mod m) means m|(a b), i.e. (a b) is divisible by m.
(b) (i) This is false as we can take b = 2, c = 3, n
1
Solutions to Final Exam of 2009
1. Question 1.
(a) 8x 6 (mod 10) is equivalent to 4x 3 (mod 5) and 3x 15 (mod 21) is
equivalent to x 5 (mod 7). Thus we are solving the following system of linear
congruences:
4x 3 (mod 5),
3x 4 (mod 11),
x 5 (mod 7).
Not
Nanyang Technological University
Semester II, 20062007
H3 Mathematics Numbers and Matrices
Final Examination
Date: 16 May 2007
Time allowed: 2 hours.
INSTRUCTIONS TO CANDIDATES
This examination paper contains SIX (6) questions and comprises TEN (10)
prin
Nanyang Technological University
SPMS/Division of Mathematical Sciences
NTU H3 Numbers and Matrices 2008
Revision Problems Test 1
1. Is the following statement true or false?
Let n be a positive integer and a is any integer. Then gcd(a, a + n)|n.
If you t
Nanyang Technological University
SPMS/Division of Mathematical Sciences
NTU H3 Numbers and Matrices 2013
Tutorial 9
Reading Determinants.
1. The following Hill 2-cipher is intercepted
LAN EQEIL
Given that the rst four plaintext letters are VANG, nd the pl
Nanyang Technological University
SPMS/Division of Mathematical Sciences
NTU H3 Numbers and Matrices 2013
Tutorial 9
Reading Determinants.
1. The following Hill 2-cipher is intercepted
LAN EQEIL
Given that the rst four plaintext letters are VANG, nd the pl
Nanyang Technological University
SPMS/Division of Mathematical Sciences
NTU H3 Numbers and Matrices 2013
Tutorial 10
Reading for this tutorial: Linear Transformations
1. A transformation T refers to a function from Rn to Rm . Here, the vector w is the ima
Nanyang Technological University
SPMS/Division of Mathematical Sciences
NTU H3 Numbers and Matrices 2013
Tutorial 10 solution
1. A transformation T refers to a function from Rn to Rm . Here, the vector w is the image
of x under T . Find the domain and cod
Nanyang Technological University
SPMS/Division of Mathematical Sciences
NTU H3 Numbers and Matrices 2013
Tutorial 11
1. A country is divided into three demographic regions. It is found that each year 5% of
the residents of region 1 move to region 2, and 5
Nanyang Technological University
SPMS/Division of Mathematical Sciences
NTU H3 Numbers and Matrices 2013
Tutorial 11 solution
1. A country is divided into three demographic regions. It is found that each year 5% of
the residents of region 1 move to region
Nanyang Technological University
SPMS/Division of Mathematical Sciences
NTU H3 Numbers and Matrices 2013
Tutorial 12
1. For each of the following, a set of objects is given together with operations and .
Determine which sets are vector spaces under the gi
Nanyang Technological University
SPMS/Division of Mathematical Sciences
NTU H3 Numbers and Matrices 2013
Tutorial 13 solutions
1. For each of the following, a set of objects is given together with operations and .
Determine which sets are vector spaces un
Nanyang Technological University
SPMS/Division of Mathematical Sciences
H3 Numbers and Matrices - 2013
Tutorial 14
Reading for this tutorial: Linear combinations, span, linear independence,
basis, dimension.
1. Which of the following sets of vectors in R3
Nanyang Technological University
SPMS/Division of Mathematical Sciences
NTU H3 Numbers and Matrices 2013
Tutorial 14 solutions
1. Which of the following sets of vectors in R3 are linearly independent ?
(a) cfw_(4, 1, 2), (4, 10, 2) (b) cfw_(3, 0, 4), (5,
Nanyang Technological University
Semester II, 2008-2009
H3 Numbers and Matrices
Test 1
INSTRUCTIONS TO CANDIDATES
Time allowed: 60 minutes.
This test paper contains FOUR (4) questions and comprises FIVE (5) printed pages.
Answer all questions. The mark
Nanyang Technological University
Semester II, 2007-2008
H3 Numbers and Matrices
Mid-Term Test 1
INSTRUCTIONS TO CANDIDATES
This test paper contains FOUR (4) questions and comprises FIVE (5) printed pages.
Answer all questions. The marks for each questio
Nanyang Technological University
Semester II, 20062007
H3 maths Numbers and Matrices
Mid term test 2
Date: 4 April 2007
Time allowed: 1 hour.
INSTRUCTIONS TO CANDIDATES
This examination paper contains FOUR (4) questions and comprises FIVE (5)
printed pag
1
Solutions to Test2 of April 4, 2007
1. Question 1.
(a) A being invertible means that there exists an n n matrix B such that BA =
AB = In , where In is the n n identity matrix. In this case, B is called an
inverse of A.
(b) Suppose A is invertible with b
Nanyang Technological University
Semester II, 20062007
H3 maths Numbers and Matrices
Mid term test 1
Date: 14 February 2007
Time allowed: 1 hour.
INSTRUCTIONS TO CANDIDATES
This examination paper contains FOUR (4) questions and comprises FIVE (5)
printed