MAT 1024 FINAL EXAMINATION 2022 mat1024

MAT1024 FINAL EXAMINATION 2022 ACADEMIC SESSION : 2022 SUBJECT : LINEAR ALGEBRA & APPLICATIONS TIME ALLOWED : 3 HOURS + 10 MINUTES READING TIME INSTRUCTIONS TO CANDIDATES There are FIVE (5) questions printed on SIX (6) pages excluding the cover page. ANSWER ALL QUESTIONS. All answers must be written on A4 foolscap paper. Write your student ID on the top right corner of each page. You are required to scan your answers using CamScanner and upload a single .pdf file on eLearn. The scanned pages must be clear and question numbers visible. You should rename your file according to the following format before submission: StudentName_StudentID.pdf IMPORTANT NOTES TO CANDIDATES Materials Allowed Standard Items: Pen, Pencil, Ruler, Eraser or Correction Fluid Special Items : Non Programmable Calculators, Zoom, Webcam, CamScanner Mobile Application Turn on and face the webcam for the entire duration of the exam. It is your responsibility to ensure that the internet browser is ONLY to be used to download the question paper and to upload your file on eLearn. Question 1 (a) (Total: 20 marks) Three positive integers, x, y, and z are given such that the sum of the average of any two integers, and the third integer, are 17, 18, and 19, respectively. Form a system of linear equations using the information provided, and solve the system by using GaussJordan elimination. (b) (6 marks) Given the following system of linear equations: ( 1024) = a 0 x y − 2022 x 0 a ( y − ) + + = (i) Explain why the system must be consistent. (ii) (2 marks) Find all the possible value(s) of a such that the system has a non-trivial solution. (4 marks) − (c) T A A n ). Suppose 1 = for a square matrix A of size n , A  find all the possible value(s) of det( (d) (4 marks) of size n n Given the following sequence of elementary row operations performed on a matrix B  . n 2 3 n n 1 → 2 R2 R R R R nR ⎯⎯⎯⎯ → B B B B → B → 2 3 3 = ⎯⎯⎯⎯→ ⎯⎯⎯⎯→ 3 Derive an expression for det(Bn ), in terms of det(B) and n. (4 marks)