Exercises for advanced program2019

b Use Cramer’s rule to solve the system of linear equations with A as the augmented matrix the last column of A is a unrestrained coefficient column.. c Find the values of m to matrix B

EXERCISES FOR AEP Lecturer: Dr Tong Thanh Trung Department of Fundamental Mathematics National Economics University

PART 1: LINEAR ALGEBRA AND ITS APPLICATIONS

1. Find the values of x and y if

2 0 2

y

x y

2 Find the values of x and y if

1 2 1

x

3. Find the values of x and y if:

2 5 7 2 5 7

4 Find the values of x and y if

1 0

x

y and

3 1

2 0

are to be equal

5. Find the values of y and z if

1 0 0

0 1 1

0 1 0

  and

1 0 1

y z

y z

are to be equal

6 For A, given below obtain 3A:

1 0 0

0 1 0

0 0 1

A

7 For the matrices given below obtain A B− and A B+ , where possible:

1 0 0

2 0

0 1 0 ,

2 1

0 0 1

(a)

0 1 0 , 1 1 1

−

(b)

8 Obtain for the row vector a and the column vector b below, the product ab and ba

a=[1 2 0], b

1 0 1

−

 

 

=  

 

9 Perform the following matrix multiplication to obtain ABwhere possible if:

4 3

1 0 0 , 1 1

0 0 1

0 2

(a)

4 3

1 0 0

B , 1 1

0 0 1

0 2

A

(b)

10 Suppose that a firm produces two types of output using three types of input Its

output quantities are given by the column vector:

q

15,000 27,000

and the prices of theses are given in the row vector p=[10 12] The amounts of

inputs it uses are given in the column vector

11,000 15,000 15,000

and the input prices are given by w=[10 10 8] Find the profit of this firm.

11 Verify that the matrix I3 below is idempotent:

3

1 0 0

0 1 0

0 0 1

I

12 Verify that the matrix A below is idempotent:

1

11

A

=

13 Verify that the matrix A below is idempotent:

1 1

A

−

14 For the matrices A and B below, verify thattrace AB( )=trace BA( ):

3 1

1 1 1

, 0 1

2 2 0

1 1

−

15 For the matrix A below, obtain trace A( ), trace AA( ), trace AAA( ):

1 6 1 3 1 6

1 3 2 3 1 3

1 6 1 3 1 6

A

−

(a)

1

11

A

=

(b)

16 Compute the determinant of the following matrices:

3 0 4

0 5 1

A

a

2 3 4

5 1 6

C

−

c

2 4 3

1 4 1

B

−

b

4 3 0 ( ) 6 5 2

9 7 3

D

d

17 Find the inverse matrix of each matrix in question 16.

18 Evaluate the determinant of B:

3 3 3

if

7

a b c

19 Find the determinant of C:

a b c

if

3

a b c

20 Find the determinant of D:

if

a b c

21 Compute

3

A

where:

1 0 1

1 1 2

1 2 1

A

22 Obtain the inverse of the following matrices by the cofactor method:

( ) 0 1 2 ( ) 0 1 0 ( ) 0 1 0

−

23 Find the inverse of the following matrices by the method of Gauss-Jordan

elimination

( ) 0 1 2 ( ) 0 2 1

−

24 Suppose that a firm produces three outputs y y1 , 2 and y3with three inputs

1 , and 2 3

z z z The input-requirements matrix is given by A below:

1 0 5

1 1 0

3 2 6

A

If the firm wants to produce 7 units of y1, 5 units of y2, and 18 units y3, how much of z z1 , and 2 z3 will be require?

25 Compute A in as few steps as possible:

3 10 6 8

A

=

26 Find the determinant of the following matrices:

1 5 6

2 7 9

A

−

a

( )

B

=

b

1 1 3 0

0 1 0 4 ( )

1 2 8 5

1 1 2 3

C

=

c

27 Evaluate the following determinants:

a) b) c)

28 Show that: (ABC)−1=C B A−1 −1 −1.

29 Given two the following matrices

4 1 2 ;

10 3 5 6

1 5 4 10

m

a) Determine the elements of second row of matrix ABT (where BT is the transpose matrix of the matrix B)

b) Use Cramer’s rule to solve the system of linear equations with A as the augmented matrix (the last column of A is a unrestrained coefficient column)

c) Find the values of m to matrix B is invertible and the element of second row, third column of matrix 4B-1 equals 6

30 Given two the following matrices

3 2 4

1 3 14 ;

m

m

−

−

a) Determine the elements of third row of matrix ABT (where BT is the transpose matrix of the matrix B)

b) Use Cramer’s rule to solve a system of linear equations with A as the augmented matrix (the last column of A is a free coefficient column) c) Find the values of m to matrix B is invertible and the element of second row, third column of matrix -3B-1 equals

3

5

31 Suppose that the demand and supply functions are numerically as follows:

10 2

2 3 15

1 2

d s d s

= − +

= − +

= + −

= − +

What will be the equilibrium solution?

32 The demand and supply functions of a two-commodity market model are

follows:

18 3 12 2

2 4 2 3

Use Crammer’s rule to find the P i and Q i (i=1,2 )

33 Given the following model:

0,0 1 : taxes 0,0 1 : income tax rate

= + +

= + − > < <

= + > < <

a) How many endogenous variables are there?

b) Find , and Y T C

34 Let the national-income model be:

( ) ( )

0

0 0, 0 1

0 1

= + +

= + − > < <

a) Identify the endogenous variables

b) Give the economic meaning of the parameter g

c) Find the equilibrium national income

d) What restriction on the parameters is needed for solution to exist?

PART TWO: CALCULUS AND ITS APPLICATIONS

1. Find the third derivative of the following functions:

3

2

1 a) ln b) 2 3 2 3 c) 3 1

2 1 d) e) ln 2 1 f) 3

4

x

x

−

−

−

2. Prove that: the function y e= +x 2e2x satisfies the following formular

y − y + y − y=

3. A firm has demand function is

1400 10 7.5 75

and total cost function is

TC Q= − Q + Q+ Find the quantities of product to maximizing

profit of the firm

4. Given the total-cost function and revenue function of the firm are

Find the value of marginal cost and marginal profit at x = 256 and interpret the economic meaning of that results

5. Given the total-cost function and revenue function of the firm are

3 2 2

Find the quantities of product to maximizing profit of the firm

6. Given the total-cost function and revenue function of the firm are

Find the quantities of product to maximizing profit of the firm

7. Given total revenue function of a monopoly manufacturer at each level of

output Q is TR=500Q- 4Q2 Find the elasticity of demand at price P = 300

and explain the economic meaning of obtained result

8. Find the quantities of product to maximizing profit of the firm, given the

marginal revenue and marginal cost functions, respectively, are

2

9. Evaluate the first partial derivatives of the following functions:

4

a) b) 5 3 c)

d) 2 4 e) ln 2 3 1 d)

x

x y

y x

10. Evaluate the first partial derivatives of the following functions:

4

a) u 3 b) u 5 3 2 c) u

d) u 2 4 e) u ln 2 3 4 d) u

xz

x yz

yz x

11. Find the total differential of the following functions

3 4 a) b) ln 3

2

x y

x y

+

−

12. Evaluate the following integrals

( )

2

2

a) ln b) c)

1

d) e) 2 3 f)

10

x

x

dx

−

− +

13. Evaluate the following definite integrals:

2

a) b) 1 c) ln 1

1

d) 4 e) f) ln

x

e x

xdx

x

+

−

14. Determine whether each improper integral is convergent or divergent, and

calculate its value if it is convergent

0

a) b) ln c)

d) e) f)

−

−∞

−∞

15. Find the relative maximum and minimum values of the following fuctions:

a) 10 6 24 b) 4 7 36

c) 3 15 12 d) 18 8 27

16. A two-product firm faces demand and cost functions below:

Q = − P P Q− = − −P P =Q + Q +

a) Find the output levels that satisfy the first-order condition for maximum profit

b) Check the second-order sufficient condition Can you conclude that this problem possesses a unique absolute maximum?

c) What is the maximum profit?

17. Find the maximum and minimum values of the following function:

w 8y= − × × −4 a x 2y 35+ Subject to the constraint 4 a x 6y 25.× × + = (With a = 1, 2, 3, 4, 5).

18. Find the maximum and minimum values of the following function:

2

z 2x= +4x 3.a.y 5+ − Subject to the constraint 4x 3.a.y 12+ = (With a = 1, 2, 3, 4, 5).

19. Find the maximum and minimum values of the following function:

z a.x 2y 1= + − Subject to the constraint x2+4y2 = +a2 1 (With a = 1, 2, 3, 4, 5).

20. Find the maximum and minimum values of the following function:

2

w 5x= +3x 4.a.y 24− + Subject to the constraint 5x 4.a.y 14+ = (With a = 1, 2, 3, 4, 5).

Exercises 15, 16, 20, 21 (Calculus and its applications, Marvin L Bittinger and David J Ellenbogen, Pages 570, 571)

Exercises 6, 7, 8, 20, 21, 22, 24, 25, 30, 31, 32, 33 (Calculus and its applications, Marvin L Bittinger and David J Ellenbogen, Pages 586, 587).