Air University
(Mid-Semester Examination: Spring 2024)
Subject:
Course Code:
Class:
Semester:
Section:
Linear Algebra
MA 201
BSDS
IV
A
Total Marks:
Date:
Time:
Duration:
FM Name:
HoD Signatures:
Note:




50
28-3-24
11:00-13:00
2 Hours
Dr. Ambreen Khan
FM Signatures:
All questions must be attempted. Solve all parts of a question together.
This examination carries 25% weight towards the final grade.
Understanding of Questions is a part of the exam, no query will be entertained.
Calculators are allowed.
Q. No. 1 (CLO 1)
20 Marks
Determine whether the following linear system is consistent or inconsistent, if it is
consistent then find the solution of the system by using Gauss Jordan Elimination
method. Also write the solution in vector parametric form.
a
b
3𝑥2 − 6𝑥3 + 6𝑥4 + 4𝑥5 = −5
3𝑥1 − 7𝑥2 + 8𝑥3 − 5𝑥4 + 8𝑥5 = 9
3𝑥1 − 9𝑥2 + 12𝑥3 − 9𝑥4 + 6𝑥5 = 15
−2 −7
Let 𝐴 = [ 2
5
1
3
the other columns.
−9
6 ]. Find the third column of 𝐴−1 without computing
4
Q. No. 2 (CLO 2)
10
10
15 Marks
Find the value(s) of 𝑘 for which the vectors {𝒗𝟏 , 𝒗𝟐 , 𝒗𝟑 } are linearly dependent. Justify
your answer.
a
1
3
−1
𝒗𝟏 = [−1] , 𝒗𝟐 = [−5] , 𝒗𝟑 = [ 5 ]
4
7
𝑘
10
Give geometric description of Span {𝒂𝟏 , 𝒂𝟐 } for the vectors.
b
8
12
𝒂𝟏 = [ 2 ] , 𝒂𝟐 = [ 3 ]
−6
−9
5
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a
Q. No. 3 (CLO 2)
15 Marks
Suppose an economy has four sectors, Agriculture (A), Energy (E), Manufacturing
(M), and Transportation (T). Sector A sells 10% of its output to E and 25% to M
and retains the rest. Sector E sells 30% of its output to A, 35% to M, and 25% to T
and retains the rest. Sector M sells 30% of its output to A, 15% to E, and 40% to T
and retains the rest. Sector T sells 20% of its output to A, 10% to E, and 30% to M
and retains the rest.
10
(i)
(ii)
b
Construct the exchange table for this economy.
Develop the system of linear equations that leads to prices at which
each sector’s income matches its expenses.
Let T: R 2  R 2 be a linear transformation. Find the standard matrix A that first
𝝅
reflect the point through vertical x2-axis and then rotates point through 𝟐 radians.
5
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