QBUS1040 Assignment 4
Semester 1, 2022
Out: 26th April 2022
Due: 11th May 2022 at 11:59pm
Instructions
• This assignment consists of seven problems. Four require a written response and three involve coding.
The problems that require a written response are described in this document, while the coding questions
are described in the associated Jupyter notebook (.ipynb file).
• You should submit a PDF to GradeScope for the written questions and match the page number with
the questions that you answered. You can find the detailed instructions on how to scan and submit
your assignments through GradeScope on Canvas. If you fail to match the page to the corresponding
question, the marker will not be able to view your response and thus you will be awarded 0 marks for the
question.
• You should answer the coding questions by modifying the Jupyter notebook appropriately, and submit it
through Canvas.
• You may not use notation, concepts or material from other classes (say, a linear algebra class you may
have taken) in your solutions. All the problems can be done using only the material from this class, and
we will deduct points from solutions that refer to outside material.
Question: 1 2 3 4 5 6 7 Total
Points: 15 10 10 10 10 10 15 80
i
QBUS1040 Assignment 4 Due 11th May 2022
1. Suppose the n×n matrices A and B are both invertible. For each case below, either (i) determine C−1, the
inverse of the matrix C; or (ii) provide an example where C is not invertible (i.e., give explicit invertible A
and B where the matrix C is not invertible).
(a) (3 points) C = B −A.
(b) (4 points) C =
[
A 0
0 B
]
.
(c) (5 points) C =
[
A 0
A−B B
]
.
(d) (3 points) C = A(A+B)B.
2. (10 points) Suppose A and D are square or wide matrices such that the product AD makes sense. We have
seen that if A and D are both right-invertible with right inverses B and E respectively, then EB is a right
inverse of AD. Now suppose B is the pseudo-inverse of A and E is the pseudo-inverse of D. Is EB the
pseudo-inverse of AD? Prove that this is always true or give an example for which it is false.
3. (10 points) Let B and D be invertible matrices of sizes m ×m and n × n, respectively, and let C be any
m× n matrix. Find the inverse of
A =
[
B C
0 D
]
in terms of B−1, C and D−1. (The matrix A is called block upper triangular.)
Hints. First get an idea of what the solution should look like by considering the case when B, C, and D
are scalars. For the matrix case, your goal is to find matrices W , X, Y , Z (in terms of B−1, C, and D−1)
that satisfy
A
[
W X
Y Z
]
= I
Use block matrix multiplication to express this as a set of four matrix equations that you can then solve.
The method you will find is sometimes called block back substitution.
4. Consider a system of linear equations Ax = b, where A is a p× n matrix, and b is a p-vector. (Therefore, x
is necessarily an n-vector.) We consider the case when p < n, i.e., A is a wide matrix and that the system
is underdetermined.
(a) (2 points) Assume that A has linearly independent rows. Show that x∗ = AT (AAT )−1b is a solution
to the system.
(b) (3 points) Show that if x is any other solution to the system, it must be of the form x = x∗ + z where
z is a vector that satisfies Ax = 0.
(c) (2 points) Let z be a vector such that Az = 0. Show that zTx∗ = 0.
(d) (3 points) Deduce that amongst all vectors x that satisfy Ax = b, x∗ is the one with the smallest
possible norm ∥x∥.
5. (10 points) Please refer to the Jupyter Notebook file for this problem.
6. (10 points) Please refer to the Jupyter Notebook file for this problem.
7. (15 points) Please refer to the Jupyter Notebook file for this problem.
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