School of Mathematics
Monash University
Semester 1 2021
ENG2005
Assignment 1
Due: Thursday, 1 April 2021, 11pm (Clayton)/ 8pm (Malaysia)
Complete the following questions, upload and submit them in Moodle in a size A4 pdf file in week 5 of the semester, no
later than Thursday 1 April, 11 pm Melbourne time (8 pm Malaysia time). Late assignments will be penalised at 10% of
the maximum mark per day late. Justify all your answers.
Learning outcomes:
• Use essential concepts related to mxn linear systems, including linear independence and basis
• Perform change of variables for multivariable functions with the chain rule, use polar coordinates
Question 1 (10 marks)


0 −1 −2
2 .
Let A be the matrix 3 4
1 1
3
(a) Find the characteristic equation of A.
(b) Find the eigenvalues of A.
(c) What are the algebraic and geometric multiplicities of the eigenvalues?
Question 2 (5 marks)
Let F (t) be a twice differentiable function of one variable. Define the function f (x, y) = yF xy for {(x, y) ∈
R2 , y 6= 0}. Show that f (x, y) satisfies
x2
2
∂2f
2∂ f
=
y
.
∂x2
∂y 2
Question 3 (10 marks)
Suppose a unit ball is located with centre 1 unit distance above the centre of another unit ball. We consider the
region in the intersection of the two balls.
(a) Sketch the region, indicating information used in the subsequent parts of the question.
(a) Express the volume of the region as an integral in both Cartesian and cylindrical coordinate systems.
(b) Calculate the volume of the region. (HINT: use the easier coordinate system!)
Question 4 (10 marks)
Locate and watch the Topic 1 video called ‘How to find the LU decomposition, and why this method works’.
The following code is an attempt to create a simple MATLAB function which captures the algorithm described
in the example in this video.
Write out this code and use the function to answer the following questions. Important: you should only submit
information that is asked of you in the question. Do not submit your MATLAB files or copies thereof.
(a) What are the three missing rows A, B and C? Write them in your submission and also include them in
your completed code to help you answer the proceeding questions (but do not submit your full code, it is
enough to submit only the missing parts A, B and C).
(b) Use the function to find the LU decomposition of a 6 × 6 matrix containing just 1’s and 0’s where 1’s
are on the leading diagonal as well as the first and third diagonals to the right as well as the third and
fourth diagonals to the left (note if you have entered the correct matrix and part (a) is correct your LU
decomposition should contain only integers). To be clear,
In your answer, just tell us what your output L and U are, no other working is needed.
(c) Using pen and paper (showing working) and your computed LU decomposition from part (b), use forward
and back substitution to find the solution vector x to Ax = b, where b = [u, v, v, u, u, v]T , in terms of
the arbitrary scalars u and v.
(d) Now use your code and attempt to find the LU decomposition of the 6 × 6 matrix formed by writing
numbers 1 to 6 on the first row, 7 to 12 on the second row, and so on.
Investigate the response of MATLAB to this input. Briefly explain in words what is going on here and
why the function fails. It may be useful to know that ‘NaN’ in MATLAB means ‘Not a Number’ and
often arises when you are attempting to divide a number by zero.