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LIA502S - LINEAR ALGEBRA 1 - 2ND OPP - JULY 2023 |
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nAml BIA UnlVERSITY
OF SCIEnCE Ano TECHnOLOGY
FACULTY OF HEALTH, NATURAL RESOURCES AND APPLIED SCIENCES
SCHOOL OF NATURAL AND APPLIED SCIENCES
DEPARTMENT OF MATHEMATICS, STATISTICS AND ACTUARIAL SCIENCE
QUALIFICATION: Bachelor of Science; Bachelor of Science in Applied Mathematics and Statistics
QUALIFICATION CODE: 07BSOC; 07BSAM LEVEL:
5
COURSE CODE:
LIA502S
COURSE CODE: LINEAR ALGEBRA 1
SESSION:
JULY 2023
PAPER:
THEORY
DURATION:
3 HOURS
MARKS:
100
SUPPLEMENTARY/
EXAMINER:
MODERATOR:
SECOND OPPORTUNITY EXAMINATION QUESTION PAPER
DR. DSI IIYAMBO
DR. N CHERE
INSTRUCTIONS
1. Attempt all the questions in the booklet provided.
2. Show clearly all the steps used in the calculations.
3. All written work must be done in black or blue inked, and sketches must be done in
pencil.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 3 PAGES (Including this front page)
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Question 1
Consider the vectors a = 2i + 2j - k and b = 2i - j + 2k.
a) Find the angle 0 (in radians) that is between a and b.
[5]
b) Find a unit vector that is perpendicular to both vectors a and b.
[7]
Question 2
Consider the following matrices.
~),B = (~~l ),
-2
2 -2
a) Given that C = AB, determine the element c32.
[3]
b) Find (3Af.
[3]
c) Is DB defined? If yes, then find it, and hence calculate tr(DB).
[6]
Question 3
Let A be a square matrix.
a) What does it mean to say that A is a skew-symmetric matrix?
[2]
b) Prove that A - AT is a skew-symmetric matrix.
[5]
c) Prove that AA T is a symmetric matrix.
[4]
Question 4
Consider the matrix B = (
2 ~4 ) .
2 3 -1
a) Use the Cofactor expansion method, expanding along the first row, to evaluate the detenni-
nant of B.
[8]
b) Is B invertible? If it is, use Gaussian reduction to find B- 1 .
[14]
c) Find det (((2B)- 1f).
[6]
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Question 5
Use Cramer's Rule to find the solution of the following system of linear equations, if it exists.
Xl + X2 + 3x3
6
x1 + 2x2 + 4x3
9
2x1 + x2 + 6x3
11
[8]
Question 6
a) Prove that in a vector space, the negative of a vector is unique.
[9]
b) Determine whether the following set is a subspace of lRn.
[13]
2