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LIA502S - LINEAR ALGEBRA 1 - 1ST OPP - JUNE 2023 |
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nAml BIA UnlVERSITY
OF SCIEnCE AnD 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:
JUNE 2023
PAPER:
THEORY
DURATION:
3 HOURS
MARKS:
100
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
EXAMINER:
DR. DSI IIYAMBO
MODERATOR:
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 p = 3i - 5j - 2k, q = i - 3j + 12k and r = i - 6k
a) Find a vector of magnitude 3 in the direction of q.
[6]
b) Find the angle (in degrees) between p and r. Give you answer correct to 1 d.p.
[8]
c) Calculate the projection of p onto r, ProjrP·
[5]
Question 2
(-~10:025) Consider the matrices A =
a) Without evaluating the whole product, determine the elements
(i) in the third row and second column of AB
[3]
(ii) in the second row and second column of BC
[3]
b) Given that atr(A) + lOtr(C) = 12, find the value(s) of a which satisfies this equation. [4]
Question 3
Let F = ( : ; :) .
-3 z 3
a) Given that the matrix Fis symmetric, give the values of x, y and z.
[5]
b) Prove that if A and B are both n x n symmetric matrices such that AB= BA, then AB is
a symmetric matrix.
[6]
c) Prove that if A is an invertible symmetric matrix, then A - l is also symmetric.
[6]
Question 4
Conside, the mat,ix A ( ~l ; ~5 ) .
a) Use the Cofactor expansion method, expanding along the second column, to evaluate the
determinant of A.
[6]
b) Is A invertible? If it is, use the adjoint method to find A- 1.
[14]
c) Find <let (3(2A)- 1).
[6]
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Question 5
Use the Gaussian elimination method to find the solution of the following system of linear
equations, if it exists.
X + 2y
2
2x
+z 1
3x + 2y + z 3
[8]
Question 6
a) Prove that in a vector space, the negative of a vector is unique.
[9]
b) Let 1'1nnbe a vector space whose elements are all then x n matrices, with the usual addition
and scalar multiplication for matrices. Determine whether the following set is a subspace
of Ivlnn·
S = {A E lVlnnI tr(A) = O}
[11]
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