linear system matrixes questions

MT/M132:Linear Algebra (KSA ONLY) Tutor-Marked Assignment (TMA) Spring 23/24 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:………………………………………………… MT/M132 / TMA Page 1 of 3 2023/2024 Spring 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) The linear system { 𝑥 − 𝑘𝑦 = 3 , has infinitely many solutions𝑘 = 4. 4𝑥 − 𝑦 = 12 b) If 𝐴 is both symmetric and skew symmetric then 𝐴 is the Zero matrix. c) If |𝐴| = |𝐴−1 |, then 𝐴 = 𝐼. d) If 𝐴𝐵 = 𝐼 then, 𝐵 = 𝐴−1 . e) The vectors 𝑣1 = (2,1), 𝑣2 = (1,2) and 𝑣3 = (5,7) are independent. Q−2: [5+5+5 marks] 2𝑥 − 5𝑦 + 5𝑧 = 17 a) Consider the linear system { 𝑥 − 2𝑦 + 3𝑧 = 9 , −𝑥 + 3𝑦 − 𝑧 = −6 (i) (ii) Solve the linear system. Find the determinant for the coefficient matrix. 1 2 0 b) Is the linear system 𝐴𝑋 = 𝐵 independent system if 𝐴 = [ 3 −1 2 ]. Justify. −2 3 −2 MT/M132 / TMA Page 2 of 3 2023/2024 Spring Q−2: [7+5+3 marks] 0 a) Find 𝑥, 𝑦 𝑎𝑛𝑑 𝑧 if 𝐴 = [𝑥 𝑥 2𝑦 𝑦 −𝑦 𝑧 −𝑧] satisfies 𝐴𝑇 = 𝐴−1 where 𝐴 is invertible. 𝑧 3 4 1 3 9 7 b) Let 𝐴 = [−1 −3 3], and 𝐵 = [1 11 7].. 2 3 0 7 5 7 (i) Find the matrix 𝐴−1 . (ii) Find a matrix 𝑋, such that 𝐴𝑇 𝑋 = 𝐵. Q−3: [8+4+3 marks] 2 0 10 −3 Let 𝐴 = [0 𝑥 + 7 −3] and 𝐵 = [ 1 ] 0 4 𝑥 0 a) Find values of 𝑥 for which the matrix 𝐴 is invertible. b) Find |3𝐴𝐴−1 | and |2𝐴2 𝐴−1 | if 𝑥 = 1. c) Is the linear system has unique solution at 𝑥 = −3. Justify. Q−4: [7+8 marks] a) Determine whether the following set of vectors in ℝ4 is linearly independent or linearly dependent. 𝑆 = {(4,2,6,0), (1,3, −2.5), (0,2, −4,6), (−1,2,0, −3)}. b) Is the vector 𝑢 = (3,3 − 8) as a linear combination of the vectors 𝑣1 = (2,1, −1), 𝑣2 = (2,0,4) and 𝑣3 = (1,2, −3). End of the questions MT/M132 / TMA Page 3 of 3 2023/2024 Spring