MAT1024 Final Exam Questions Linear Algebra and Applications Apr 2023.

MAT1024 Final Exam
Questions Linear Algebra
and Applications Apr 2023.

, School of Mathematical Sciences MAT1024 / July 2022



Question 1 (Total: 20 marks)

(a) 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 Gauss-
Jordan elimination.
(6 marks)

(b) Given the following system of linear equations:

( a −1024 ) x + 2022 y = 0
2022 x + ( a −1024 ) y = 0

(i) Explain why the system must be consistent.
(2 marks)

(ii) Find all the possible value(s) of a such that the system has a non-trivial
solution.
(4 marks)

(c) Suppose AT = A−1 for a square matrix A of size n  n, find all the possible value(s) of
det( A).
(4 marks)

(d) Given the following sequence of elementary row operations performed on a matrix B
of size n  n.
R2 → 2 R2
B = B1 ⎯⎯⎯⎯ → ⎯⎯⎯⎯B2 → R3 → 3R3 B3 Rn → nRn
⎯⎯⎯⎯→ Bn


Derive an expression for det( Bn ), in terms of det( B) and n.
(4 marks)




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