The American University in Cairo
Department of Economics
ECON 316-02 (Mathematics for Economists II)
Instructor: Diaa Noureldin
Fall 2013
Graded Assignment I
Due date: October 7, 2013, 10:00AM
(to be submitted in class)
1. Use the following matrices to answer the questions below:
2
3
2
3
2
3
6 1
7
6
7
3 1
3 2
4
5 ; D = 2;
5; B = 6 2 9
A = 4
3 7
4
5; C =
1 2
5 4
4 2
4
2
3
2 3
2
3
1 8 3 7
6
7
6 7
6
3
1
5
6 6 9 6 3 7
6 3 7
6
7
7
6 7
7; F = 6
;
K
=
E = 6
6
7
6 7;
2
4
5
4
5
6 9 7
6 2 4 9 7 7
4
5
4 5
2
7
12
4 8 5 2
4
2
3
2
3
9 3
6
7
5 5 3
7
4
5:
L = 6
4 2 4 5; M =
6 9 4
1 6
Find (if feasible):
(a) AC.
(b) BF .
(c) BL.
(d) DF .
(e) LM .
(f) DA
C.
(g) L(A + C).
(h) jEj.
1
(i) jM j.
(j) Show that (F K)0 = K 0 F 0 .
2. Let Xm
n
be any matrix, and let P = X(X 0 X)
1X 0.
Show that P is an idempotent matrix.
3. Three Interrelated Markets: Consider the markets for co¤ee (c), tea (t) and (s). Note
that the …rst two goods are substitutes, while the third is often complementary with both
co¤ee and tea. The following is the demand and supply equations for all three markets:
Q(d)
= 120
c
Q(s)
=
c
= 100
(s)
=
Qt
2Ps
20 + 5Pc
(d)
Qt
8Pc + 2Pt
5Pt + 3Pc
Ps
10 + 2Pt
Qs(d) = 300
Ps
5Pc
10Pt
Q(s)
= 15Ps :
s
Assuming that the markets are in equilibrium, express the linear system of equations as
2
3
Pc
6
7
7
Ax = b; where x = 6
4 Pt 5
Ps
is the vector equilibrium prices.
4. Leontief Input-Output Model: An economy consists of three sectors: agriculture, industry
and services. To produce $1 worth of agricultural output, the agricultural sector requires $0.3
of its own output, $0.1 of industrial output and $0.4 of services output. On the other hand,
to produce $1 worth of industrial output, the industrial sector requires $0.4 of agricultural
output, $0.3 of its own output and $0.2 of services output. Finally, to produce $1 worth
of services output, the services sector requires $0.1 of agricultural output, $0.4 of industrial
output and $0.4 of its own output.
(a) Derive the input requirements matrix A from the above information.
3
2
35; 000
7
6
7
(b) Given that the vector of output x = 6
4 50; 000 5, …nd the vector of …nal demand d in
40; 000
the economy.
2
5. The following is a system of three equations in three unknowns (x; y; z):
x =
2y
3z + 4
y = z+1
z =
x
2y + 2:
(a) Use matrix algebra to express the system as Ax = b where
2
x
3
6 7
7
x=6
4 y 5;
z
(b) Solve for x using Cramer’s rule.
6. Find the inverse of the following two matrices:
2
2
6
A=6
4 3
1
1
2
2
1
3
7
4 7
5;
3
2
6
B=6
4
2
4
1
1 1
3
7
2 2 7
5:
2 3
7. Using two di¤erent methods, …nd the eigenvalues of the following matrix:
2
3
3 1
5:
A=4
2 4
8. For the matrix
2
A=4
2 1
1 2
3
5;
(a) Write the characteristic equation and …nd the characteristic roots.
(b) Find its eigenvectors q1 and q2 .
h
i
(c) Create the matrix Q = q1 q2 and show that it is an orthogonal matrix.
3