TMA solve

Plagiarism Warning:

As per AOU rules and regulations, all students are required to submit their own TMA work and avoid plagiarism. The AOU has implemented sophisticated techniques for plagiarism detection. You must provide all references in case you use and quote another person’s work in your TMA. You will be penalized for any act of plagiarism as per the AOU’s rules and regulations.

Declaration of No Plagiarism by Student (to be signed and submitted by student with TMA work):

M/MT132: Linear Algebra
KSA – TMA Fall 24/25
Cut-Off Date: Based on the Published Deadline.
Total Marks: …. marks turned to 15 marks
Contents
Warnings and Declaration…………………………………….……………………………………1
Question 1 ……………….…………………………………. ……………………………………..2
Question 2 ………………………………………………………………………………….…..…..3
Question 3 ………………………………………………………………………………….…..…..4
Question 4 ………………………………………………………………………………….…..…..5
Marking Criteria ……………..………………………………………………………….………..…6
Plagiarism Warning:
As per AOU rules and regulations, all students are required to submit their own TMA
work and avoid plagiarism. The AOU has implemented sophisticated techniques for
plagiarism detection. You must provide all references in case you use and quote
another person’s work in your TMA. You will be penalized for any act of plagiarism as
per the AOU’s rules and regulations.
Declaration of No Plagiarism by Student (to be signed and submitted by student
with TMA work):
I hereby declare that this submitted TMA work is a result of my own efforts and I have
not plagiarized any other person’s work. I have provided all references of information
that I have used and quoted in my TMA work.
Name of Student:……………………………..
Signature:……………………………………………
Date:…………………………………………………
M/MT132 / TMA
Page 1 of 3
2024/2025 Fall
The TMA covers only chapters 1 and 2. It consists of four questions; each question
is worth 15 marks. You should give the details of your solutions and not just the final
results.
Q−1: [5×3 marks]
Answer each of the following as True or False (1 mark) justifying your answers (2
marks):
a) If 𝐴𝐵 = 𝐼 then, 𝐵𝐴 = 𝐼.
b) The linear system {
3𝑥 + 𝑘𝑦 = 9
, has infinitely many solutions 𝑘 = −3.
2𝑥 − 2𝑦 = 6
c) The vectors 𝑣1 = (−1,0,3), 𝑣2 = (2, −3,1) and 𝑣3 = (4,0,2) are dependent.
d) Any matrix 𝐴 can be written as the sum of symmetric and skew symmetric matrix.
e) If 𝐴 is a nonsingular square matrix such that 𝐴2 = 𝐼 then, 𝐴−1 = 𝐼.
Q−2: [5+3+3+4 marks]
𝑥+𝑦+𝑧 =2
Consider the linear system {𝑥 + 3𝑦 + 3𝑧 = 0,
𝑥 + 3𝑦 − 6𝑧 = 3
a) Find the values of 𝑎, 𝑏, 𝑎𝑛𝑑 𝑐 if the augmented matrix of the linear system is in row
1 2 3 𝑎
equivalence to the matrix [0 1 2 |𝑏 ].
0 0 1 𝑐
b) Solve the linear system.
c) Is the coefficient matrix 𝐴 singular? Explain.
d) Find |2𝐴𝑇 (𝐴2 )−1 |.
M/MT132 / TMA
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2024/2025 Fall
Q−3: [7+3+5 marks]
−𝑚1 𝑥 + 𝑦 = 𝑏1
a) Consider the linear system {
,
−𝑚2 𝑥 + 𝑦 = 𝑏2
i)
ii)
Prove that if 𝑚1 ≠ 𝑚2 , then the linear system has a unique solution?
Suppose that 𝑚1 = 𝑚2 . Then under what conditions will the linear system be
consistent?
1
1 3
b) Let 𝐴 = [
]. Find 𝑐 𝑎𝑛𝑑 𝑑 such that 𝐴2 = 𝟎.
𝑐 𝑑
c) Prove that if 𝐴 is an 𝑛 × 𝑛 matrix and it satisfies the equation 𝐴3 − 4𝐴2 + 3𝐴 −
5𝐼𝑛 = 𝟎, then 𝐴 is nonsingular. Find its inverse.
Q−4: [7+8 marks]
a) Determine whether the following set of vectors in ℝ3 is linearly independent or
linearly dependent, where 𝑆 = {(3,0, −1), (2,1, −4), (0,1,2)}.
b) Is the vector 𝑢 = (2, −5, −1,6) as a linear combination of the vectors
𝑣1 = (3, −1,0,1), 𝑣2 = (1,2,0, −3) and 𝑣3 = (−1,3, −1,5).
End of the questions
M/MT132 / TMA
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2024/2025 Fall