8002A Semester 1 2012

The University of Sydney

School of Mathematics and Statistics

MATH1002

Linear Algebra

June 2012 Lecturers: A. Crisp, R. Crossman, H. Dullin, R. Howlett

Time Allowed: One and a half hours

Family Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Other Names: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

SID: . . . . . . . . . . . . . Seat Number: . . . . . . . . . . . . .

This examination has two sections: Multiple Choice and Extended Answer.

The Multiple Choice Section is worth 50% of the total examination;

there are 25 questions; the questions are of equal value;

all questions may be attempted.

Answers to the Multiple Choice questions must be entered on

the Multiple Choice Answer Sheet.

The Extended Answer Section is worth 50% of the total examination;

there are 3 questions; the questions are of equal value;

all questions may be attempted;

working must be shown.

Students may bring and use their own approved non-programmable

calculators

THE QUESTION PAPER MUST NOT BE REMOVED FROM THE

EXAMINATION ROOM.

Marker’s use

only

Page 1 of 32

8002A Semester 1 2012 Page 2 of 32

Multiple Choice Section

In each question, choose at most one option.

Your answers must be entered on the Multiple Choice Answer Sheet.

1. What is the magnitude of i 2j 2k?

(a)

p

5 (b) 2 (c) 3 (d) 5 (e) 9

2. If u = i+ 2k and v = i+ 2j then what is u · v?

(a) 3 (b) 2j+ 2k (c) 2 (d) 0 (e) 1

3. If a = i+ j 3k and b = i+ 2k then what is a⇥ b?

(a) 2i 5j k (b) 5 (c) 2i 5j+ k (d) i+ j+ 4k (e) 2i j+ k

4. What is the cosine of the angle between the vectors 2i+ j+ 2k and 3i 4k?

(a) 1 (b) 4

15

(c) 2

15

(d)

2

15

(e)

1

2

5. If b⇥ c = i and a⇥ b = 2j+ k, then what is (a b+ c)⇥ b?

(a) i+ 2j+ k (b) i+ j 2k (c) i 2j+ k (d) i+ j k (e) 2j+ k

8002A Semester 1 2012 Page 3 of 32

This blank page may be used for rough working; it will not be marked.

Be sure to enter your answers on the Multiple Choice Answer Sheet.

8002A Semester 1 2012 Page 4 of 32

6. What is the product AB of the matrices A =

1 2

1 2

and B =

4 6

2 3

?

(a)

2 3

4 6

(b)

0 0

0 0

(c)

10 5

20 10

(d)

16 8

16 8

(e)

0 3

4 0

7. What is the determinant of

24 2 1 21 3 1

1 0 1

35?

(a) 1 (b) 10 (c) 27 (d) -10 (e) 0

8. The matrix

24 1 0 1 3 20 1 1 1 3

2 0 2 6 0

35 is the augmented matrix of a system of linear equations.

Which one of the following statements is true?

(a) The system has five unknowns.

(b) The system is inconsistent.

(c) The system has a unique solution.

(d) The general solution has one parameter.

(e) The general solution has two parameters.

9. Let B be a 3⇥ 3 matrix and suppose that det(B) = 1 . Then what is det(2B)?

(a) 8 (b) 2 (c) 1/8 (d) 2 (e) 8

8002A Semester 1 2012 Page 5 of 32

This blank page may be used for rough working; it will not be marked.

Be sure to enter your answers on the Multiple Choice Answer Sheet.

8002A Semester 1 2012 Page 6 of 32

10. The invertible n⇥ n matrices A, B and C are such that A1B = B1CB.

Which expression equals the matrix A?

(a) C1B (b) BC1 (c) B1C (d) CBC (e) B1CB1

11. If A, B and C are 4⇥ 4 matrices, and BAC = 2I4, then what is the inverse of A?

(a) 2B1C1 (b) 1

2

BC (c)

1

2

CB (d) 1

2

CB (e) BC

12. Which one of the following planes contains the point with position vector i j k?

(a) x+ 2y + 2z = 5

(b) 2x y z = 0

(c) (r+ j) · (i+ j+ 2k) = 1

(d) r · (i j k) = 0

(e) 3x y 2z = 6

13. What is the area of the triangle with vertices P (1, 0, 0), Q(0,1, 0), R(0, 0, 2)?

(a) 3/2 (b) 2 (c) 3 (d) 9/2 (e) 5

8002A Semester 1 2012 Page 7 of 32

This blank page may be used for rough working; it will not be marked.

Be sure to enter your answers on the Multiple Choice Answer Sheet.

8002A Semester 1 2012 Page 8 of 32

14. Which one of the following vectors is perpendicular to the line

x 2

3

= y 1 = z + 1

2

?

(a) i+ j+ 3k (b) 2i 3j (c) 3i+ j+ 2k

(d) i j k (e) 2i j+ k

15. Let v = j and w = i+ j+ k. The vector projection of v in the direction of w is

(a) j. (b) 1. (c) 1p

3

(i+ j+ k).

(d) i+ k. (e) 13(i+ j+ k).

16. Consider the following system of linear equations:

x + y + z + w = 0

x + y w = 0

x + z = 0

Which one of the following statements about this system is true?

(a) There is no solution.

(b) There is a unique solution.

(c) The general solution is expressed using 1 parameter.

(d) The general solution is expressed using 2 parameters.

(e) The general solution is expressed using 3 parameters.

17. Consider the following system of equations:

3x y + z = 3

x + y + 2z = 1

Which one of the following is not a solution to this system?

(a) x = 1 3t, y = 5t, z = 4t, where t 2 R

(b) x = 1 + 3t, y = 5t, z = 4t, where t 2 R

(c) x = 3t, y = 53 + 5t, z = 43 4t, where t 2 R

(d) x = 3t, y = 5t, z = 4t where t 2 R

(e) x = 5, y = 10, z = 8

8002A Semester 1 2012 Page 9 of 32

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Be sure to enter your answers on the Multiple Choice Answer Sheet.

8002A Semester 1 2012 Page 10 of 32

18. Consider the following system of equations where a is a constant:

x y 2z = 1

x + y + z = 0

2x 2y z = a

Which one of the following statements about this system is true?

(a) The system is always consistent.

(b) The system is consistent if and only if a = 1.

(c) If a = 1 then the system has a unique solution.

(d) If a = 1 then the system has infinitely many solutions expressed using 1 parameter.

(e) If a = 1 then the system has a unique solution.

19. Which one of the following statements may be false for some square matrices A, B and

C of the same size, with I standing for the identity matrix?

(a) A = B implies AC = BC

(b) A(B C)A = ACA+ ABA

(c) (A B)2 = A2 2AB +B2

(d) (A+ I)(B I) = AB A+B I

(e) (I + A)(I A) = I A2

20. The two lines given by the respective parametric equations

x = t

y = 1 t

z = 1 + 2t

9=; t 2 R and x = 1 sy = sz = 1 2s

9=; s 2 R

(a) are the same line. (b) are parallel to each other.

(c) intersect at the point (1, 0, 1). (d) intersect at the point (1, 1, 2).

(e) do not intersect.

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This blank page may be used for rough working; it will not be marked.

Be sure to enter your answers on the Multiple Choice Answer Sheet.

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21. The following two lines

r = i+ j+ k+ t(2i 2j 2k), t 2 R

and

x 1 = y 2

2

=

z + 1

1

intersect each other. What is the equation of the line (where s 2 R) passing through

the intersection point of these two lines and perpendicular to both of them?

(a) r = i+ j+ k+ s(i j k)

(b) r = i+ 2j+ 3k+ s(i 2j+ 7k)

(c) x = y23 =

z1

2

(d) r = s(i+ k)

(e) r = j k+ s(i k)

22. Given the matrix A =

0 1

1 0

, what is A4?

(a) [1] (b)

0 1

1 0

(c)

1 0

0 1

(d)

0 1

1 0

(e)

1 0

0 1

23. What are the eigenvalues of the matrix

24 1 20 10 2 0

1 12 1

35?

(a) 0, 0, 2

(b) 2,p2,p2

(c) 2, 0, 0

(d) 2, i,i

(e) 2,1, 1

8002A Semester 1 2012 Page 13 of 32

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Be sure to enter your answers on the Multiple Choice Answer Sheet.

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24. Which one of the following statements about the matrix

1 2

0 1

is true?

(a) 1 is an eigenvalue with eigenspace

n0

t

t 2 Ro.

(b) 1 is an eigenvalue with eigenspace

n t

t

t 2 Ro.

(c) 1 is an eigenvalue with eigenspace

nt

2t

t 2 Ro.

(d) 1 is an eigenvalue with eigenspace

n t

2t

t 2 Ro.

(e) 0 is an eigenvalue with eigenspace

n2t

t

t 2 Ro.

25. If A = P DP1 where D and P are the matrices D =

2 0

0 3

and P =

1 1

1 0

,

then what is A7?

(a)

27 37

27 0

(b)

37 27 37

0 27

(c)

27 0

0 37

(d)

27 37 37

27 0

(e)

2185 2059

0 128

This space may be used for rough working; it will not be marked.

8002A Semester 1 2012 Page 15 of 32

End of Multiple Choice Section

Make sure that your answers are entered on the Multiple Choice Answer Sheet

The Extended Answer Section begins on the next page

8002A Semester 1 2012 Page 16 of 32

Extended Answer Section

There are three questions in this section, each with a number of parts. Write your answers

in the space provided below each part. If you need more space there are extra pages at the

end of the examination paper.

1. (12 marks in total)

(a) Let u and v be the vectors u = 2i+ j 6k and v = 2i k.

Resolve u into a sum of two vectors, one parallel to v and the other perpendicular

to v. (2 marks)

Question 1 continues on the next page

8002A Semester 1 2012 Page 17 of 32

(b) Let a and b be nonzero vectors that are perpendicular to each other.

(i) Use the geometric definition of the dot product to show that a · b = 0.

(1 mark)

(ii) Show that if µ 2 R is any scalar then the vectors c = a+ µb and d = a µb

have the same length. (1 mark)

(iii) Find the value of µ2 if the vectors c and d in Part (ii) are also perpendicular.

(1 mark)

Question 1 continues on the next page

8002A Semester 1 2012 Page 18 of 32

(c) Consider the following system of linear equations

x 2z = 2

x y + z = 1

2x+ y 4z = 2

where 2 R is a parameter.

(i) Write down the augmented matrix for the system and use Elementary Row

Operations to reduce the matrix to Row Echelon Form. Describe the opera-

tions that you are performing at each stage. (3 marks)

Question 1(c) continues on the next page

8002A Semester 1 2012 Page 19 of 32

Use the final augmented matrix from Part (i) to answer the following questions:

(ii) Show that the system of equations is inconsistent when = 4. (1 mark)

(iii) Find the value of for which the system has infinitely many solutions, and

find the corresponding one dimensional parametric solution. (2 marks)

(iv) For what values of does the system have a unique solution? (1 mark)

Question 2 begins on page 21

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Question 2 begins on the next page

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2. (12 marks in total)

(a) Let E =

24 2 1 21 a 1

2 0 1

35 and F =

241 1 b1 2 c

2 2 1

35, where a, b and c are integers.

(i) Calculate (in terms of a) the determinant, det(E), of the matrix E. (1 mark)

(ii) Calculate det(3E1). (1 mark)

(iii) If E = F1, find a, b and c. (3 marks)

Question 2 continues on the next page

8002A Semester 1 2012 Page 22 of 32

(b) Show that if all the entries of a matrix M and of its inverse M1 are integers then

det(M) = ±1. [Hint: use the property that det(MM1) = det(M) det(M1).]

(3 marks)

Question 2 continues on the next page

8002A Semester 1 2012 Page 23 of 32

(c) Let A =

24 2 3 22 0 1

1 1 1

35. You are given that A1 =

24 1 1 31 0 2

2 1 6

35 .

Make use of these matrices to solve the following systems of equations.

(i) x+ y + 3z = 4

x 2z = 1

2x y 6z = 0 (2 marks)

(ii) 2x+ z = 2

3x+ 3y 3z = 9

2x+ 3y 2z = 0 (2 marks)

Question 3 begins on page 25

8002A Semester 1 2012 Page 24 of 32

Question 3 begins on the next page

8002A Semester 1 2012 Page 25 of 32

3. (12 marks in total)

(a) Let B =

240 0 40 1 0

3 4 4

35, and let v1 =

24 1615

4

35 and v2 =

2420

3

35.

(i) Use matrix multiplication to show that v1 and v2 are eigenvectors of B and

find the corresponding eigenvalues 1 and 2. (2 marks)

(ii) Calculate the characteristic polynomial det(B I), verify that 1 and 2

from Part (i) are roots, and find the third eigenvalue 3. (2 marks)

(iii) Find an eigenvector v3 corresponding to 3. (1 mark)

(iv) Write down matrices P and D such that B = PDP1 and D is a diagonal

matrix. (You are not required to find P1.) (2 marks)

Question 3 continues on page 27

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Question 3 continues on the next page

8002A Semester 1 2012 Page 27 of 32

(b) Use vectors to show that any angle inscribed in a semicircle is a right angle.

(3 marks)

•• •

•

O

P R

Q

Question 3 continues on the next page

8002A Semester 1 2012 Page 28 of 32

(c) Let A, B and C be three points that do not lie on the same straight line. (That is,

they are not collinear.)

Let a =

!

OA, b =

!

OB and c =

!

OC be the position vectors of these points relative

to the origin O.

Show that the vector

n = a⇥ b+ b⇥ c+ c⇥ a

is normal to the plane that passes through A, B and C. (2 marks)

There are no more questions.

Extra blank pages are provided in case you need more space for your answers.

8002A Semester 1 2012 Page 29 of 32

This blank page may be used if you need more space for your answers

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This blank page may be used if you need more space for your answers

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This blank page may be used if you need more space for your answers

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This blank page may be used if you need more space for your answers

End of Extended Answer Section

This is the last page of the question paper.

8002B Semester 1 2012 Multiple Choice Answer Sheet

0 0

⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

1 1

⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

2 2

⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

3 3

⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

4 4

⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

5 5

⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

6 6

⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

7 7

⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

8 8

⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

9 9

⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Write your

SID here !

Code your

SID into

the columns

below each

digit, by

filling in the

appropriate

oval.

Answers !

a b c d e a b c d e

Q1 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q2 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q3 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q4 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q5 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q6 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q7 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q8 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q9 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q10 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q11 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q12 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q13 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q14 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q15 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q16 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q17 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q18 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q19 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q20 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q21 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q22 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q23 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q24 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

Q25 ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥ ⇤⇥

The University of Sydney

School of Mathematics and

Statistics

MATH1002 Linear Algebra

Family Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Other Names: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Seat Number: . . . . . . . . . . . . . . . . .

Indicate your answer to each question by

filling in the appropriate oval.

This is the first and last page of this answer sheet