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LIA502S - LINEAR ALGEBRA 1 - 2ND OPP - JAN 2023 |
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nAml BIA UntVERSITY
OF SCIEnCE AnDTECHnOLOGY
FACULTY OF HEALTH, APPLIED SCIENCES AND NATURAL RESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of Science in Applied Mathematics and Statistics
QUALIFICATION CODE: 07BSAM
LEVEL:
5
COURSE CODE:
LIA502S
COURSE CODE: LINEAR ALGEBRA 1
SESSION:
JANUARY 2023 PAPER:
THEORY
DURATION:
3 HOURS
MARKS:
100
SUPPLEMENTARY/
EXAMINER:
MODERATOR:
SECOND OPPORTUNITY EXAMINATION QUESTION PAPER
MR. GS MBOKOMA, DR. N CHERE
DR. DSI IIYAMBO
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 Jone 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
1.1 State whether each of the following statements is true or false. Justf:f'y yov.r answer.
a) If a, b and c are any three vectors in JR3; , then a• (b + c) = a x b + a x c.
[2]
b) j x i = k.
[3]
c) If AB and BA are both defined, then A and B are square matrices.
[3]
d) If matrix A has a column of all zeros, then so does AB if this product is defined. [3]
e) The expressions tr(AT A) and tr(AAT) are defined for every matrix A.
[2]
f) The sum of two diagonal matrices of the same size is also a diagonal matrix.
[3]
1.2 Given that u = ni + 5j - v'7kand iul = 9, find the possihle values of the scalar O'..
[4]
1.3 Determine the area of parallelogram whose adjacents sides are a = 2i - 4j + 5k and b =
i - 2j - 3k. Leave your answer iu surd form.
[5]
Question 2
2.1 Write down a 4 x 4 matrix whose if" entry is given by aij = ij~l , and comment on your
matrix.
[6]
2.2 Let A be a square matrix. State what is meant by each of the following statements.
a) A is symmetric
[2]
b) A is orthogonal
[2]
c) A is skew-symmetric
[2]
2.3 Consider the following matrices.
~), 1 -?
A= 4 2
B= (
( 0 1 -2
-2
~l),and D=
2
(21
2
1
3)_
4
a) Given that C = AB, determine the element c32 .
[4]
b) Find (3Af.
[3]
c) Is DB defined? If yes, then find it, au<l 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]
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Question 3
Consider the matrix B = ( ~2 ~4 ) .
2 3 -1
a) Is B invertible? If it is, use the Gauss-Jordan Elimination method to find B- 1 .
[12]
Question 4
Use the Crammer's rv.le to solve the following system of linear equations, if it exists.
2x - y + 3z
2
3x + y- 2z
0
2x - 2y + z
8
[8]
Question 5
a) Prove that in a vector space, the negative of each vector is unique.
[9]
b) Determine whether the following set is a subspace of IR3.
[12]
2