### Question Description

10 questions in total.

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Surname: 1

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Question 1

We describe a permutation matrix as a square binary matrix that has exactly one entry of 1 in

each column row and each row and 0s elsewhere

a) the product of permutation matrices is a permutation matrix as shown below

0

[

1

(0 ∗ 0) + (1 ∗ 1)

1

0 1

]∗[

]=[

(1 ∗ 0) + (0 ∗ 1)

0

1 0

(0 ∗ 1) + (1 ∗ 0)

0 1

]=[

]

(1 ∗ 1) + (0 ∗ 0)

1 0

a) the inverse of a permutation matrix is the transpose of the matrix as shown here:

0

[

1

=[

1

𝑎

]∗[

0

𝑐

1

𝑏

]=[

0

𝑑

0

]

1

(0 ∗ 𝑎) + (1 ∗ 𝑏) (0 ∗ 𝑏) + (1 ∗ 𝑑)

𝑏

]=[

(1 ∗ 𝑎) + (0 ∗ 𝑐) (1 ∗ 𝑏) + (0 ∗ 𝑑)

𝑎

0

Therefore b=1, d=0, a=0, c=1 which is the transpose of [

1

1

𝑑

]=[

0

𝑐

0

]

1

1

]

0

Question 2

We re given that that g, r, and C are known.

𝑟

The first thing is to make x the subject of the formula g’ x = r to obtain x=𝑔′

Also, the condition that the portfolio weights sum to one can be expressed as

Where 1 is a 3 × 1 vector with each element equal to 1. Consider another portfolio with weights

y = (y y yc)’,

Surname: 2

The first three elements of z

are the portfolio weights m = ( )0 for the global minimum variance portfolio with

expected return P,M = m0μ and variance 2

= m0Σm

Question 3

A linear system with unique solution has a solution set with one element. A linear system

with no solutions has an empty set of solutions

Ax=0 is an homogeneous equation since it has a constant of zero so that it can be written as

𝑎1 𝑥1 + 𝑎2 ...

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