ECON104 Mathematical Economics
Homework 1
Due: Online 5pm 7th Feb 2025 (extended due to CNY)
Question 1:
1. Find the scalar λ such that Ax = λx with
−2 −1 4
A= 2
1 −2
−1 −1 3
2. Find the numbers x, y, z, w such that
!
!
2 1
x y
−
−1 0
z 0
x y
z 0
and
1
x = 0
1
!
!
0 1
2 1
=
!
w −2
0 1
3. A “square” matrix A is said to be idempotent if A2 = AA = A.
• Prove that the following matrix is idempotent.
2 −2 −4
4
−1 3
1 −2 −3
• Show that if AB=A and BA=B then both A and B are idempotent.
Question 2:
0 0 0
1. Given A = 4 0 0
10 5 0
• Find B = A2 and A3 to illustrate that AB = 0 does not lead to either A = 0 or
B=0
• Find I + A + A2 and (I − A)(I + A + A2 ) where I is the identity matrix of order 3.
1
• Use the result in above part to find (I − A)−1
2. Let A = I − X XT X
−1
XT where I is an identity matrix.
• Must X be square? Must XT X be square? Must A be square?
• Show that A is idempotent i.e. AA = A
Question 3:
1. In the 2D-space, given 2 points A(3, 5), represented by vector a(3, 5), and B(1, 4), represented by vector b(1, 4).
(a) Find the length of vector a and b as well as the distance between A and B.
(b) Find the line that passes both points A and B.
• Assume the line has the formula y = αx + β and hence find α, β.
• A random point X(x1 , x2 ) (represented by vector x(x1 , x2 )) lies on the line that
pass AB, express coordinates of X using vector algebra: x = (1 − t)a + tb
i. What is the point X when t=0? t=1? t=-1? t=3?
ii. Verify that all the points in part a) above belong to the lines in part i)
2. There are 2 commodities with price vector: p = (2, 5) and the budget for the consumer
is $50.
(a) Assume that the consumer uses x units of good 1 and y units of good 2. What
does (x,y) have to satisfy so that it is an affordable bundle? Represent y in terms
of x. What is it?
(b) Prove that the line obtained in part a) is orthogonal to the price vector (hint:
take 2 random points in the budget line and check its inner product with the price
vector).
(c) Write the budget line in vector format if given one possible bundle of consumption
is (15, 4).
2
0
