The University of Sydney
School of Mathematics and Statistics
Assignment 2
MATH1002: Linear Algebra Semester 1, 2023
Lecturers: Bregje Pauwels, Ruibin Zhang
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Copyright © 2023 The University of Sydney 1
1. Let M =
1 −2 12 1 −3
4 −3 −1
, x0 =
13
5
, and [M | x0] =
1 −2 1 12 1 −3 3
4 −3 −1 5
.
(a) Find the reduced row echelon form of the augmented matrix
[
M | x0
]
.
(b) Using the result of part (a), or otherwise, solve the following system of linear
equations
M
xy
z
 = x0 (1)
to find all solutions.
(c) i). Show that if v0 and v1 are distinct solutions of equation (1) in part (b), then
v = v1 − v0 satisfies
Mv = 0;
ii). Use i) and results of part (b) to find an explicit eigenvector of M with eigen-
value 0.
2. Let T =
−2 1 11 2 −3
0 0 3
, and denote I3 =
1 0 00 1 0
0 0 1
.
(a) i). Calculate det (T − xI3) for any x to find the characteristic polynomial of T ;
ii). Determine the eigenvalues of T .
(b) Find an eigenvector of T associated with each of the eigenvalues of T .
(c) Express an arbitrary vector v =
ab
c
 in R3, where a, b, c ∈ R, as a linear combi-
nation of eigenvectors of T .
3. Let ω =
[
0 1
−1 0
]
, and let A be any 2× 2 matrix over R with detA = 1.
(a) Show that ATωA = ω.
(b) Use part (a), or otherwise, show that A−1 = ω−1ATω.
(c) We define u ∗ v = uTωv for any u,v in R2. Show that for all x,y ∈ R2,
i). x ∗ y = −y ∗ x; and
ii). (Ax) ∗ (Ay) = x ∗ y.