MTH2021 Linear Algebra with Applications
MTH2025 Linear Algebra (Advanced)
Assignment 1
School of Mathematics, Monash University
This assignment is worth 6% of your total mark for this unit. Your completed assignment must be submitted via the corresponding Moodle
Assignment activity by 5pm, April 1st. A single .pdf file must be submitted. No cover sheet need be attached. Late assignments may be
accepted, however they will generally incur a late penalty. Late assignments should be submitted via Moodle. Problems marked with a † treat
material only covered in MTH2025 and should be ignored by MTH2021 students. MTH2025 students should attempt all problems.
1. (a) Let A be anm⇥ n matrix, and let B be the result of performing a single elementary row operation on A. Prove that if A 6= O then
B 6= O.
[(2 marks, MTH2021) , (1 mark, MTH2025)]
(b) Let A be an m ⇥ n matrix, and let B be the result of performing a single elementary row operation on A, and let w 2 Rn. Prove
that if every row of A is a multiple of w, then every row of B is also a multiple of w.
[(2 marks, MTH2021) , (1 mark, MTH2025)]
(c) Let A be an m ⇥ n matrix. Prove that if the reduced echelon form of A possesses precisely one pivot, then there exist non-zero
vectors v 2 Rm and w 2 Rn such that A = vwT .
[(2 marks, MTH2021) , (2 marks, MTH2025)]
2. Let G = (V,E) be a directed graph with V = {1, 2, . . . , n} and with adjacency matrix A such that (A)ij  1 for all i, j 2 V . LetM be
the n⇥ n matrix such that (M)ij is the number of vertices to which both i and j are connected by edges, i.e.
(M)ij = |{k 2 V : ik 2 E} \ {k 2 V : jk 2 E}|
Find an explicit expression forM as a function of A.
[(4 marks, MTH2021) , (3 marks, MTH2025)]
3. Consider an economy with six product-producing sectors and one non-producing sector, described by the following table, where the
input is in dollars needed for $1.00 of output.
Input Required per Dollar Output
Pr
ov
id
er
Widgets Gizmoes Doohickies Thingamabobs Wotsamajigies Hoosywotsits
Widgets 1/6 1/7 1/8 1/9 1/10 1/11
Gizmoes 1/6 1/7 1/8 1/9 1/10 1/11
Doohickies 1/6 1/7 1/8 1/9 1/10 1/11
Thingamabobs 1/6 1/8 1/7 1/10 1/9 1/11
Wotsamajigies 1/6 1/8 1/7 1/10 1/9 1/11
Hoosywotsits 1/6 1/8 1/7 1/10 1/9 1/11
In answering the following questions, you are free to use Mathematica to perform any necessary calculations. However, your written
answers must clearly explain the underlying concepts, and you must meaningfully interpret your results.
(a) Is the consumption matrix corresponding to this economy productive?
[(2 marks, MTH2021) , (1 mark, MTH2025)]
(b) Suppose the open sector consumes $2w of widgets, $3w of gizmoes, $2w of doohickies, $w of thingamabobs, $w of wotsamajigies
and $2w of hoosywotsits. What dollar output of doohickies must the economy produce in order to satisfy demand?
[(2 marks, MTH2021) , (1 mark, MTH2025)]
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,
4. (a) Let n 2 and let x1, x2, . . . , xn+1 2 R with xi 6= x1 for all 2  i  n + 1. Define n ⇥ n matrices A and B so that for all
1  i, j  n we have (A)ij = xj1i+1 and
(B)ij =
(
xji1 , i  j
0, i > j
Prove that for all 1  i, j  n
(AB)ij =
xji+1 xj1
xi+1 x1
Hint: You may use the elementary fact that for all x, y 2 R andm 2 N
xm ym = (x y)
mX
k=1
xk1ymk
[(1 mark, MTH2021) , (1 mark, MTH2025)]
(b) Let n 2 and let x1, x2, . . . , xn+1 2 R with xi 6= x1 for all 2  i  n+ 1. Define the (n+ 1)⇥ (n+ 1) matrixM so that for all
1  i, j  n+ 1 we have (M)ij = xj1i . Prove that
det(M) =

nY
k=1
(xk+1 x1)
!
det(AB),
where A and B are as defined in Part (a).
Hint: You may find it useful to consider the Laplace expansion of M along its first column, after first performing some relevant
row operations. You may also find Part (a) useful.
[(2 marks, MTH2021) , (2 marks, MTH2025)]
(c) LetM and A be defined as above. Prove that
det(M) =

nY
i=1
(xi+1 x1)
!
det(A).
[(1 mark, MTH2021) , (1 mark, MTH2025)]
(d) Consider the following proposition: For any choice of n 2 and any choice of x1, x2, . . . , xn 2 R, if P denotes the n⇥ n matrix
with entries (P )ij = xi1j for 1  i, j  n, then det(P ) =
Q
1iHint: You may find Part (c) useful.
[(2 marks, MTH2021) , (2 marks, MTH2025)]
5.† A square matrix P is stochastic if each entry of P is non-negative, and the entries in each column of P sum to 1. Let Pn denote the set
of all n ⇥ n stochastic matrices. Does Pn form a group under matrix multiplication? Explain exactly which group axioms Pn satisfies,
and which (if any) it does not.
[(0 marks, MTH2021) , (2 marks, MTH2025)]
6.† Consider the symmetric group Sn.
(a) What is the maximum number of inversions a permutation can have, and which permutation(s) realise(s) this maximal number?
[(0 marks, MTH2021) , (2 marks, MTH2025)]
(b) How many adjacent transpositions does Sn contain?
[(0 marks, MTH2021) , (1 mark, MTH2025)]
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