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MATHEMATICAL METHODS FOR ENGINEERS 2 and LINEAR ALGEBRA
Assignment 2 – due Thurs 5th Sept, 4pm
Please use a cover sheet and submit (together with the assignment questions from tutorial 4)
via the MME2 (or Linear Algebra, if appropriate) assignment box in the OC building. Make
sure you include your tutor’s name and tutorial day/time on your cover sheet.
Late submission incurs a penalty of 10% per day, up to 40%. Work more than 4 days late will
not be accepted.
1. Consider the linear system of equations for (x, y, z) which can be represented by the
following augmented matrix
!
”
”
”
#
2 !2 4
… !4
1 0 4
… 1
2 !2 m ! 1
… n ! 3
$
%
%
%
&
(a) Perform 2 or more row operations to reduce the system to a point where it can be
solved. You do not need to reduce the system to reduced row echelon form.
(b) Find the values of m and n for which the system has
i. No solutions
ii. Infinitely many solutions
iii. A unique solution
(c) In the case in which the system has infinitely many solutions, find the solution in
terms of a parameter, t.
Tutorial 3. Week 4
1. Suppose matrix A is invertible and you exchange its first two rows to obtain matrix B.
Is the new matrix B invertible? If yes, how would you find B!1 from A!1?
2. Prove that a matrix with a column of zeros is not invertible.
3. Let A =
‘
(
)
!1 2 1
1 4 !1
!1 2 2
*
+
,
(a) Use RREF to find A!1. Hence find the solution to linear system AX = B if B =
(2 ! 3 1)T .
(b) Write A!1 as a product of elementary matrices.
(c) Hence write A as a product of elementary matrices.
4. Given A =
‘
(
)
a b c
0 d e
0 0 f
*
+
,
where a, b, c, d, e, andf are parameters, specify restrictions on
the values of parameters, so that the inverse A!1 will exist.
MATHEMATICAL METHODS FOR ENGINEERS 2 and LINEAR ALGEBRA
Assignment 2 – due Thurs 5th Sept, 4pm
Please use a cover sheet and submit (together with the assignment questions from tutorial 3)
via the MME2 (or Linear Algebra, if appropriate) assignment box in the OC building. Make
sure you include your tutor’s name and tutorial day/time on your cover sheet.
Late submission incurs a penalty of 10% per day, up to 40%. Work more than 4 days late will
not be accepted.
2. Consider the matrix A =
!
”
#
0 0 !2
1 2 1
1 0 3
$
%
&
.
(a) By performing a suitable matrix multiplication, show that X1 and X2 are eigenvec-
tors of A, where X1 = [!2, 1, 1]
t and X2 = [!1, 0, 1]
t. What are the corresponding
eigenvalues?
(b) Using your answer to (a), find the third eigenvalue and its corresponding eigenvector.
Choose any constants so that X3 has simple entries and it is linearly independent to
X1 and X2.
(c) Is A diagonalisable? Write down a matrix P which will diagonalise A and write
down the corresponding diagonal matrix D.
3. This question will be similar to Q5 from Tutorial 4. It will be given to you in your tutorial
during Week 6. It should be completed and submitted to your tutor at the end of the
allocated time. The relevant section of the formula sheet on the course website will be
provided – no other notes are allowed, no calculators allowed. Note that each tute
class will have a slightly di!erent question.
Tutorial 4. Week 5
1. Let A =
‘
1 4
3 5
(
. By performing matrix multiplications, determine which of the follow-
ing are eigenvectors of A:
‘
2
1
(
,
‘
2
!1
(
,
‘
3
2
(
,
‘
2
3
(
.
What are the corresponding eigenvalues? Check that their sum and product match the
trace and determinant of A.
2. Find eigenvalues of the matrix by solving its characteristic equation.
(a)
‘
!1 2
3 4
(
(b)
‘
3 1
0 3
(
Additional questions: Kleinfeld and Kleinfeld, Section 3.1.1, p. 81, Questions (part
(a) only) 2, 4, 6.
