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LIA502S - LINEAR ALGEBRA 1 - 2ND OPP - JANUARY 2024 |
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I, '
.~
n Am I BIA u ni VE Rs ITY
OF SCIEnCEAnDTECHnOLOGY
FacultyofHealth,Natural
ResourceasndApplied
Sciences
Schoolof NaturalandApplied
Sciences
Departmentof Mathematics.
Statisticsand ActuarialScience
13JacksonKaujeuaStreet
PrivateBag13388
Windhoek
NAMIBIA
T: +264612072913
E: msas@nust.na
W: www.nust.na
QUALIFICATION : BACHELOR of SCIENCE IN APPLIED MATHEMATICS AND STATISTICS
QUALIFICATION CODE: 07BSAM
LEVEL:5
COURSE: LINEAR ALGEBRA 1
COURSECODE: LIA502S
DATE: JANUARY 2024
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
SECOND OPPORTUNITY/ SUPPLEMENTARY: EXAMINATION QUESTION PAPER
EXAMINER:
MODERATOR:
MR GABRIEL S MBOKOMA, DR NEGA CHERE
DR DAVID IIYAMBO
INSTRUCTIONS:
1. Answer all questions on the separate answer sheet.
2. Please write neatly and legibly.
3. Do not use the left side margin of the exam paper. This must be allowed for the examiner.
4. No books, notes and other additional aids are allowed.
5. Mark all answers clearly with their respective question numbers.
PERMISSIBLE MATERIALS:
1. Non-Programmable Calculator
This paper consists of 3 pages including this front page.
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Question 1
Consider the vectors p = i + j - 2k and q = i - 3j + 12k
a) Find the unit vector in the direction of p.
[4]
b) Find the angle (in degrees) between p and q. Give you answer correct to 1 d.p.
[8]
Question 2
2.1 Write down a 4 x 4 matrix whose i/h entry is giveu by aij = ij~l,
matrix.
and comrncut on your
[6]
2.2 Let A be a square matrix. State what ili meant by each of the following statement:;.
a) A is symmetric
[2]
b) A is orthogonal
[2]
c) A is skew-symmetric
[2]
2.3 Consider the following matrices.
3) 1 -2
A= 4 2 1 ,
( 0 1 -2
B = ( ~1),
-2 2
a) Given that C = AB, determine the element c:;2-
[5]
b) Find (3Af.
[5]
c) Is DB Jefined? If yes, then find it, and hence calculate tr(DB).
[6]
2.4 Suppose A is a square matrix. Check if the matrix B = :3(A- AT) is skew-symmetric? [5]
Question 3
Consider
3.1 Let
B=[ l· sin 0 cos 0
-cos0 sin0
Show that the det(B) = l.
[4]
3.2 Consider
~i A=[~
3 -1 4
U~e the arljoint matrix of A to fin<l .4- 1.
[10]
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Question 4
Use the Crammer's rnle to solve the following :;ystem of linear equations, if it exists.
2x - y + 3z
2
3x + y - 2z
0
2x - 2y + z
8
Question 5
a) Prove that in a vector space, the negative of each vector is unique.
b) Determine whether the following set is a subspace of JR~.
S={(x,y,z)ElR 3 lx+y+z=O}
c) Prove that if x and y are orthogonal vectors in lRn, then show that
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