The Australian National University 2018-04-30

EMET7001 Introduction to economic models

2018S1

Problem set #09

Introduction The first set of problems are for discussion in the tutorials in week 10. The problems on the second page are the homework assignment which is part of the course assessment.

Problem 1 Compute the matrix products:

.. (c).

c

Problem 2 Use Gaussian elimination to find the values of a for which the equation system

x + y - 2z = a + 7

3x -y + az = -3

-x + ay - 4z = 8

has (i) exactly one solution; (ii) more than one solution; (iii) no solution.

Problem 3 The trace, tr(A), of any n × n matrix A = (aij) is defined by tr( . That is, tr(A) is the sum of all the diagonal elements of A. Show that if A and B are n × n matrices and c is any scalar, then:

(a) tr(A + B) = tr(A) + tr(B). (b) tr(cA) = ctr(A).

(c) tr(A0) = tr(A). (d) tr(AB) = tr(BA).

Problem 4 Let a , and c . Find a + b + c, a- 2b + 2c, and 3a + 2b- 3c.

Problem 5 For what values of x is the inner product of 2x, x, and x, -x, x equal to 0?

Problem 6 Let a , and c . Compute kak, kbk and kck, and verify that the Cauchy-Schwarz inequality holds for a and b.

Problem 7 Let a .

1. Show that a is a point in the plane -x + 2y + 3z = 1.

2. Find the equation for the normal at a to the plane in part (1).

EMET7001 2018-04-30 2

Instructions The problems relate mostly to material from chapter 15 in the textbook.

• The homework is due 4 p.m. on Wednesday 9 May.

• Please write your name and U number at the top of all pages.

• Please ensure that multiple pages are securely stabled together.

• Please submit in the assignment box near the RSE reception in the Arndt building.

• Please write your tutorial time on the front page.

• Show your workings!

Problem A Compute A2 when

A

Problem B Find the unique matrix Y that satisfies the equation

Y .

Problem C Suppose that A and B are n×n matrices that satisfy A2B = AB. Prove that:

(i) A3B = AB;

(ii) A4B = AB;

(iii) AkB = AB for all integers k = 1.

Problem D

(a) Use Gaussian elimination to find the values of p and q for which the equation system

x1 + x2 + x3 = q

px1 + x2 -x3 = 5

x1 -x3 = p

has (i) one solution; (ii) several solutions; (iii) no solution.

(b) Find an expression for the general solution of the system in case (ii).

Problem E Check which of the following pairs of vectors are orthogonal:

and .

and .

(c) x, y, 0 and y, .

EMET7001 Introduction to economic models

2018S1

Problem set #09

Introduction The first set of problems are for discussion in the tutorials in week 10. The problems on the second page are the homework assignment which is part of the course assessment.

Problem 1 Compute the matrix products:

.. (c).

c

Problem 2 Use Gaussian elimination to find the values of a for which the equation system

x + y - 2z = a + 7

3x -y + az = -3

-x + ay - 4z = 8

has (i) exactly one solution; (ii) more than one solution; (iii) no solution.

Problem 3 The trace, tr(A), of any n × n matrix A = (aij) is defined by tr( . That is, tr(A) is the sum of all the diagonal elements of A. Show that if A and B are n × n matrices and c is any scalar, then:

(a) tr(A + B) = tr(A) + tr(B). (b) tr(cA) = ctr(A).

(c) tr(A0) = tr(A). (d) tr(AB) = tr(BA).

Problem 4 Let a , and c . Find a + b + c, a- 2b + 2c, and 3a + 2b- 3c.

Problem 5 For what values of x is the inner product of 2x, x, and x, -x, x equal to 0?

Problem 6 Let a , and c . Compute kak, kbk and kck, and verify that the Cauchy-Schwarz inequality holds for a and b.

Problem 7 Let a .

1. Show that a is a point in the plane -x + 2y + 3z = 1.

2. Find the equation for the normal at a to the plane in part (1).

EMET7001 2018-04-30 2

Instructions The problems relate mostly to material from chapter 15 in the textbook.

• The homework is due 4 p.m. on Wednesday 9 May.

• Please write your name and U number at the top of all pages.

• Please ensure that multiple pages are securely stabled together.

• Please submit in the assignment box near the RSE reception in the Arndt building.

• Please write your tutorial time on the front page.

• Show your workings!

Problem A Compute A2 when

A

Problem B Find the unique matrix Y that satisfies the equation

Y .

Problem C Suppose that A and B are n×n matrices that satisfy A2B = AB. Prove that:

(i) A3B = AB;

(ii) A4B = AB;

(iii) AkB = AB for all integers k = 1.

Problem D

(a) Use Gaussian elimination to find the values of p and q for which the equation system

x1 + x2 + x3 = q

px1 + x2 -x3 = 5

x1 -x3 = p

has (i) one solution; (ii) several solutions; (iii) no solution.

(b) Find an expression for the general solution of the system in case (ii).

Problem E Check which of the following pairs of vectors are orthogonal:

and .

and .

(c) x, y, 0 and y, .

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