New Semester
Started
Get
50% OFF
Study Help!
--h --m --s
Claim Now
Question Answers
Textbooks
Find textbooks, questions and answers
Oops, something went wrong!
Change your search query and then try again
S
Books
FREE
Study Help
Expert Questions
Accounting
General Management
Mathematics
Finance
Organizational Behaviour
Law
Physics
Operating System
Management Leadership
Sociology
Programming
Marketing
Database
Computer Network
Economics
Textbooks Solutions
Accounting
Managerial Accounting
Management Leadership
Cost Accounting
Statistics
Business Law
Corporate Finance
Finance
Economics
Auditing
Tutors
Online Tutors
Find a Tutor
Hire a Tutor
Become a Tutor
AI Tutor
AI Study Planner
NEW
Sell Books
Search
Search
Sign In
Register
study help
business
project management processes methodologies and economics
Mathematics For Economics 3rd Edition Michael Hoy, John Livernois - Solutions
11. Find two ways of determining whether a matrix A is positive semidefinite.
10. Find two ways of determining whether a matrix A is negative definite.
9. What is a quadratic form?
8. What is an eigenvalue?
7. What is the rank of a matrix?
6. What does orthogonality mean in terms of the geometry of vectors?
5. How would you decide if two vectors were orthogonal?
4. How does the idea of linear dependence as presented in this chapter relate to the idea of linear dependence discussed in chapter 7?
3. What is the effect on a vector of multiplying it by a negative fraction?
2. What is the relationship between the inner product of a vector and the Euclidean norm?
1. How does the idea of distance between vectors relate to the Euclidean norm?
9. Find the rank of 0 [10 1 A = 0 10 100 B=|0 1 0 1 0 10 1 1 00 1
8. If A is a 6 x 9 matrix, what is the maximum number of linearly independent columns that A may have?
7. Suppose is the positive quadrant in xy-plane defined as = {[] such that x 20 y 20 (a) If two vectors u and v are in V, is u+v in V? (b) If u is in V, is Au in V, where is any scalar? On the basis of (a) and (b), is V a vector space? 0}
6. Express the following vectors in terms of the orthonormal basis (e1, e2, e3): [1/2] (a) e 0 (b) y= 1 (c) z= 0 0 -2
5. Check whether the following pairs of vectors are orthogonal: (a) y = (b) y = 0 W (c) y = 7-8--17
4. Express the following vectors in terms of the basis vectors given by sets (a) and (b) in question 3. 5. 1 2 (a) y (b) z 2 0 5
3. Show that the following sets of vectors are linearly independent: 0 0 0 (a) e= 0 e2 = 3 0 1 0 0 0 2 0 (b) v = 0 V2= V3 = 0 0 1 0 0 0 4 4 4. Express the following vectors in terms of the basis vectors given by sets (a) and (b) in question 3. 5. 1 2 (a) y (b) z 2 0 5 Check whether the following
2. Find the inner product of the following pairs of vectors: 3. (a) y=0 w= 0 0 (b) y = [1/2 (c) y = W 1/2 0
1. Find the lengths of the following vectors: [1/3] -4-4-4-8 y 2 W = 0 v=1/3 1/3
9. Suppose that A is a square matrix such that |A| = 0. Show that A is not invertible.
8. Let A and B be square matrices, with B invertible. Show that | BAB|=A
7. Let A be a square matrix such that A7 A = 1. Show that |A|= 1.
6. Let A and B be square matrices. Show that even though AB may not be equal to BA, it is always true that |AB| = |BA|. 7.
5. Use Cramer's rule to compute the solutions to 2x1 + x2 -3x1 3 + X3 -8 X2 + 2x3 -2
4. Suppose that a firm produces three outputs, y1, y2, and y3, with three inputs, Z1, Z2, and z3. The input-requirement matrix is given by A: HNW l 2 5 1 1 3 If the firm wants to produce 10 units of m, 20 units of y2, and 10 units of y3, how much of z], z2, and 23 will it require? 5. Use Cramer's
3. Use the cofactor expansion method to compute |A|, where 1 5 0 A: 2 4 —1 0 —2 0
2. Verify that det A = det B + det C, where an 6112 “1 + u1 C111 a12 u] A = a21 a22 142 + 1)2 B = a2] ‘122 M2 6131 a32 “3 + ”3 a31 G32 143 011 (112 1)1 C = a21 a22 U2 a3] 6132 Us What do you conclude about the statement that in general for all matrices B and C, det(B + C): detB + detC?
1. Does the following matrix have an inverse? 12 13 14 A = 15 16 17 18 19 20
6. If you had a system of 10 equations with 10 endogenous variables to be de- termined, but you were only interested in the values of two of the variables, would you solve the system by (a) matrix inversion (b) Cramer's rule. Or, is it difficult to choose between the methods? Explain.
5. How is the adjoint matrix related to the cofactor matrix?
4. What is a cofactor matrix?
3. What are the main properties of determinants?
2. What is the determinant of a matrix?
1. What is the scalar equivalent of the matrix operation of multiplication by the inverse of a matrix?
8. Use the data in example 8.11 to find the unemployment rate after two periods. [Hint: The unemployment rate is the number of people unemployed as a proportion of all labor market participants.] Can the situation described by these transition probabilities evolve in the same way indefinitely?
7. Use the data of example 8.10 to find the evolution of the population after four periods.
6. Compute the quantities below using 1 2 8 4 A = [ _ } } } ] B = [ 1 } ] -} -3 4 (a) AT, BT, AT + BT, (A + B)T (b) AB. (AB), AT BT, BT AT -7 5
5. Let A = -81-84 B = [3, 4]
4. Let 1 [2 0 0 A = 1 2 3 D= 0 3 0 1 45 0 04 Compute AD and DA.
3. Let -]] A = B = 4 -[1-8] A = Compute (Ax), x A", xx, and xx. Is Ax defined?
2. Show that ABBA where
1. For each of the matrix operations below, identify those that result in a scalar. (a) AB where A is 2 x 1 and B is 1 x 2 2. (b) ATB where A is 2 1 and B is 2 1 (c) AT BA where A is 2 x 1 and B is 2 2 (d) AAB where A is 5 x 1 and B is 5 x 5
5. What is the trace of a matrix? 1. For each of the matrix operations below, identify those that result in a scalar. (a) AB where A is 2 x 1 and B is 1 x 2 2. (b) ATB where A is 2 1 and B is 2 1 (c) AT BA where A is 2 x 1 and B is 2 2 (d) AAB where A is 5 x 1 and B is 5 x 5 Show that ABBA where
4. What is the transpose of a symmetric matrix?
3. Why is it important to distinguish between premultiplication and postmulti- plication of matrices, but not for scalars?
2. When are two matrices conformable for multiplication?
1. When are two matrices conformable for addition (subtraction)?
8. Use the data in example 8.11 to find the unemployment rate after two periods. [Hint: The unemployment rate is the number of people unemployed as a proportion of all labor market participants.] Can the situation described by these transition probabilities evolve in the same way indefinitely?
7. Use the data of example 8.10 to find the evolution of the population after four periods.
6. Compute the quantities below using A -8 -[41]-[33] (a) AT, BT, AT + BT, (A+B)T (b) AB. (AB), AT BT, BT AT -7 5
5. Let A -31-1 B = 5 k What values of k, if any, will make AB = BA?
4. Let 1 200 A 1 23 D 0 3 0 1 45 004 Compute AD and DA.
3. Let A B = [49] A -[44]-8 Compute (Ax), xT AT, xx, and xx. Is Ax defined?
2. Show that ABBA where
1. For each of the matrix operations below, identify those that result in a scalar. (a) AB where A is 2 x 1 and B is 1 x 2 (b) ATB where A is 2 x 1 and B is 2 x 1 (c) AT BA where A is 2 x 1 and B is 2 2 (d) AATB where A is 5 x 1 and B is 5 x 5
5. What is the trace of a matrix?
4. What is the transpose of a symmetric matrix?
3. Why is it important to distinguish between premultiplication and postmulti- plication of matrices, but not for scalars?
2. When are two matrices conformable for multiplication?
1. When are two matrices conformable for addition (subtraction)?
5. For the matrix A below, obtain trace(A), trace(AA), and trace(AAA): 1/6 -1/3 1/6 (a) A = A = -1/3 2/3 -1/3 1/6 -1/3 1/6 6 -2 -5 -2 8 -2 -4 (b) A = || -5 -2 6 1 1 -4 1 2
4. For the matrices A and B below, verify that trace(AB) = trace(BA): 5. 3 -1 A B 0 2
3. Verify that the matrix A below is idempotent: -[AA] =
2. Verify that matrix A below is idempotent: 6 -2 -5 1 -2 8-2-4 A = 11 -5-2 6 1 -4 2
1. Verify that the matrix 3 below is idempotent: 1 0 0 --[ ] 13 = 010 001
7. How many rows does B have if BA is a 2 x 6 matrix?
6. Let B = [ } } ], b=[3-3] =[{ _32] 3 6 Verify that AB AC even though BC.
5. If the order of matrix A is 3 x 5 and that of the product AB is 3 x 7, what is the order of B?
4. Obtain the profit function of a firm that produces three types of output using three inputs. The output vector is given by 2,000 q=3,000 6,000 the price per unit of output vector is given by the input vector is given by 10 p= 15 20 [2,000 z=2,500 2,000 and the price per unit of input vector is
3. Verify that for the matrices A and B below (AB) BTAT. (a) (b) A = [ 0 0 1 ] . B = 1 1 02 23 A 00 +-- B = 210 30 1
2. For the matrices below, indicate whether the operations listed under (a)-(e) are well defined. If not, explain why. 7 0 A = 2 3 -] [ 1 3 32 =[ ]. 4 --- 0 E = 2 (a) -3A (b) A + E (c) B-3D (d) 3C-E (e) AC
1. Find the transpose of the following matrices: (a) (b) 10 0 13 = 0 1 0 00 1 A 2 3 210 30 1
6. Use equation (8.2) and the data for P and x given in example 8.10 to find the regional population distribution after three time periods.
5. 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 these are given in the row vector p = [10 12]. The amounts of inputs it uses are given in the column vector 11.0001 z= 15,000 15.000
4. Perform the following matrix multiplications to obtain AB where possible: if 4 3 (a) A =[ 100 B = 1 1 0 0 1 02 4 3 (b) B= 0 00 A = 1 1 02
3. Obtain for the row vector a and the column vectorb, below, the products ab and ba: b = a [120], b
2. A = 00 010 0 0 1 For the matrices given below obtain A - B and A + B, where possible: (a) A A = 10 01 010 B = 00 [3] - 00 2 0 (b) A 0 10 B = 1 -1 1 0 0 1 0 0-1
1. For A given below obtain 3A:
5. Find the values of y and z if 00 0 1 1 and 0 Z 0 1 0 1 Z are to be equal.
4. Find the values of x and y if 257 are to be equal. x [ ] and 2 []
3. Find the values of x and y if 3 21-61 834-830 ][ 2 5
2 Find the values of x and y if
1. Find the values of x and y if 2. [4,]-[61] x-y
5. An economy has an IS curve given by R = 210-2Y and an LM curve given by R-M+Y/4. The long-run equilibrium level of output must equal 100. What value of M makes the IS and LM curves intersect at Y = 100? What is the economic interpretation of a situation in which M exceeds this critical amount?
4. An economy has three markets with supply and demand functions for the three goods given by q = -20+p: -0.5p2 92 = -100+2p2 9 = P3 q 80-2p1 P3 = 9=200-P2 9-100-2p3 P (a) Comment on the relationship between the three goods on the demand side. (b) What is the nature of any production externality on
3. Which of the following systems are linearly dependent and which are incon- sistent? (a) 2x + y 2 = 10 4y+2z = 4 (b) x=0 -y+z = 0 4x+2y 2/3 0 x+2=0 (c) -3x+2y-z = 14 0 -x y z x+10y 3z 2
2. (a) ya+b/x a,b>0 (b) ya+b a,b>0 (c) y=x+ex (d) y=x+b b>0 (e) y Inz+ax a>0 Solve the following pairs of equations by substitution and by elimination. (a) y = 24-x 2y = 4+5x (b) y=-8x -4 y = 20x + 2 (c) 0.5y+2x = 0 -y+x=0 (d) 0.5y+2x = 0 -y-4x=0
1. Which of the following equations are linear in .x?
8. What is the distinguishing feature of a homogeneous system of linear equa tions?
7. What can be said about the appearance of an array (or matrix) that is in reduced row-echelon form?
6. Use Venn diagrams to show the relationship between consistent systems. inconsistent systems, overdetermined systems, and underdetermined systems.
5. What is meant by a consistent system of linear equations?
4. What information is conveyed by comparing the number of independent linear equations and the number of unknowns?
3. What is meant by linear independence in a system of equations?
2. What is meant by linear dependence in a system of equations?
1. Why must a system of two linear equations have either no solution, exactly one solution, or infinitely many solutions?
5. Solve the following system of excess demand functions for P1, P2, P3, P4: 10 P 2pz +5p3 2p4 = 0 - 56 6 + 2pi 20 - 2P - 2pi + P2 - P3 + 8p4 0 P2 - 4p3 - 9P4 = 0 - 2pz - 2p3 2p4 = = 0
4. A small, open economy with a flexible exchange rate has IS, LM, and BP curves given by RR 240 -50+ - 4Y + E Y R 105 + Y - 2E Solve, using row operations for the equilibrium interest rate R, output Y, and exchange rate E. (The exchange rate is defined as the price of domestic currency in terms of
3. Write the following systems in array form and derive the reduced row-echelon form in each case. Then find the solutions to each set of equations. (a) 2x 0 - x2 x2 + x3 + 2x4 = 100 x1 + 2x2 + 2x3 60 -x1 + x3 - X4 -10(b) x1 + x2 + X3 = 0 x2 = 20 -x1 + 2x3 = 10
Showing 500 - 600
of 5106
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Last
Step by Step Answers
Get 50% OFF
Claim Now
