1    Section A - Complete all problems

Each question is worth 5 marks and will be graded based on correctness of solution, explanation, and understanding demonstrated.

The additional Quality of explanation and use of notation mark will be assessed against the following standards:

Mark   Description                                                                                                              

5       Exceptional

4       Clear explanations and fluent use of notation

2-3     Overall comprehensible explanations and mostly free of notation errors

1       Lacking in clarity, detail, or significant misunderstandings with regard to notation

0       Very little shown in terms of explanation

1. Linear independence and rank

Consider 4 column vectors:                                and        , with x,y,z ∈ R.

i. Identify values of x,y,z such that the four vectors are linearly independent. Be sure to provide reasoning (e.g., along the lines of the statement in 4.1.4 of the Study Guide) and demonstrate that linear independence is satisfied.

ii. Identify values of x,y,z such that the four vectors are linearly dependent, but any three of the vectors taken together are linearly independent.

iii. In part (ii), showing the rank of the 4 × 4 matrix formed from your vectors to be 3 would not generally be a sufficient argument. Why not?

5 marks

2. Calculating an inverse matrix

Consider the following system Ax = b:

「(l)1   2(0)   1 x =  「(l)−10(5)

i. Calculate the cofactors Cij  for each entry in the matrix A and hence find adj(A).

ii. Calculate the determinant of A and hence use A−1  to solve for x.                            5 marks

3. Systems of Linear Equations

For the following system of equations,

x1 + 2x2 − x3     =   5

x2 + 3x3     =   −2

−x1 − 6x2 − 11x3     =   3

i. Represent the system as an augmented matrix, and then solve the system using the following Steps:

Step 1 - Use multiples of Row 1 to eliminate the x3  entries in rows 2, 3 and 4. Step 2 - Use Row 2 to eliminate the x2  entry in rows 3 and 4.

Step 3 - Set x1  as the free variable and then express each of x2 ,x3  in terms of x1 .

ii. Set x1  to any integer in [2, 5] and then to any integer in [−5, −2] and verify that this results in a valid solution to the system using matrix multiplication.

iii. Justify the existence of unique/infinite solutions using the concept of matrix rank.

5 marks

4. Gaussian Elimination

Use Gaussian Elimination to reduce the following system of equations to row-echelon form, hence solve for x1 ,x2 ,x3 ,x4 .

2x1 − 5x2            + 2x4     =   −3

8x1 − 20x2 + 3x3               =   −21

−4x1 + 10x2 + 3x3 − 4x4     =   1

5 marks

5. Gauss-Jordan Elimination

Consider the following matrix, A:

A =  l」

「3   −1   1

i. Use the Gauss-Jordan elimination method (Starting with the augmented matrix [A|I]) to find the inverse A−1 .

ii. Hence find the matrix B such that AB =  l」

「9     2  .

5 marks

i. Calculate A「(l)  and A「(l) , then comment on whether or not these vectors are eigenvectors of A. If they are, give the corresponding eigenvalues.

ii. Find the characteristic equation using the form

D(λ) = b3 λ3 + b2 λ2 + b1 λ + b0

and calculating the coefficients b0 ,b1 ,b2 ,b3  using the formulas on page 17 of the study guide (Topic 6 – and e.g., see online module 6.3. b2 = −tr(A) in this case).

iii.  Now identify the characteristic equation D(λ) in factorised form by first expanding the determinant |λI − A|. (Hint: you should be able to factor out one of the (λ − aii) factors first, and then expand and factorise the resulting quadratic equation)

iv. Use the factorised form of D(λ) to identify the eigenvalues of A, and then solve (λI−A)x = 0 to find all eigenvectors not identified in part (i).

5 marks

7. Matrix Similarity

i. For the matrices A and B provided below, based solely on the determinants and characteristic equations, comment on whether or not it might be possible that a matrix P exists such that P−1 AP = B . (You do not need to find the matrix P)

A = [1(3)

5(3)] ,

B = [0(3)

2]

ii. The following two matrices are similar. Find the matrices P and P −1 such that P−1 AP = B

A = [2(2)   4(0)] ,        B = ]

iii. Give one further example of a matrix C that is similar to A and B .

5 marks

8. Diagonalisation

Consider the matrix

A =   6(14)  1

−12

−10

2

J

3  

i. Determine the characteristic equation D(λ) for A.

ii. Show that D(−1) = 0 and use this information to factorise the characteristic equation using polynomial division and find the remaining eigenvalues.  Comment on whether or not we can tell if A is diagonalisable at this stage.

iii. Find the corresponding eigenvectors.

iv.  Construct P from the eigenvectors, then find P −1  (either using Gauss-Jordan elimination or the cofactors method) and then check P−1 AP to verify the correctness of your solutions.

5 marks

2    Section B - Choose 3 out of 5

Choose 3 of the following problems (if you complete more than 3, only the first three appearing in your work will be assessed). Each question is worth 5 marks and will be graded based on correctness of solution, explanation, and understanding demonstrated.

The Quality of explanation and use of notation mark will be assessed against the same standards as for Section A:

Mark   Description                                                                                                              

5       Exceptional

4       Clear explanations and fluent use of notation

2-3     Overall comprehensible explanations and mostly free of notation errors

1       Lacking in clarity, detail, or significant misunderstandings with regard to notation

0       Very little shown in terms of explanation

1. Orthogonal Matrices

i.  Identify a set of values a,b,c such that the columns of the following matrix are mutually orthogonal.

iii. Use your answer in (i) to determine an orthogonal matrix B and verify that B⊤ = B−1

5 marks

2. Writing Eigenvector Examples

Using random matrices for practicing eigenvector and eigenvalue questions will often result in non-integer values that are confusing and tedious to calculate.  Use the following to reverse engineer a question.

i.  Give three simple, linearly independent column vectors x1 , x2 , x3  ∈ R3 .  Ensure that they each have at least 2 non-zero entries, and that one of your values is negative.  (these are your eigenvectors)

ii.  Pick three eigenvalues, λ 1 ,λ2 ,λ3 .  Use integer values, and have one of your λi  such that λi  < 0.

iii.  We know that Ax = λx for each of your eigenvectors.  Hence set up a system AX = B where X = [x1x2x3] is the matrix with your eigenvectors as columns, and solve for A.

3. Balancing Traffic Flow

Consider the directed graph below showing hourly traffic in and out of 4 intersections.  The variables x1 ,x2 ,x3 ,x4  indicate the amount of traffic travelling between intersections.

200

100

1

D

4

C

x3

300

200

i.   For each vertex, we assume that the flow of traffic into the vertex (along edges directly connected to the vertex only) is equal to the flow of traffic away from the vertex.  Write an equation modeling the flow of traffic for each vertex in terms of the variables x1 ,x2 ,x3 ,x4 .

ii. Based on the four equations from (i), solve the system. Interpret your solution in context.

4. LU form of matrices

i. Use the LU decomposition method to find L and U for the following matrix (without using software or a graphics a calculator).

l 」

=

「

Note: Sometimes it is easier to calculate L and U without re-writing partial matrices as separate steps. You can set out your working, e.g. as:

and then just provide some annotations below as you fill out the entries. enough working to demonstrate that you can follow the process.

ii.   After you have LU , calculate the determinant using determinant property (ii) of 3.2.4 in the Study Guide)

You just need to show

properties  (i.e., using

l 2(4) 」                                                          l2(4) 」

iii.  Suppose Ax =  「3 .  Solve for x by first solving Ly =  「3  for y and then using this vector to solve Ux = y for x.

5 marks

5. Triangularisation with an orthogonal matrix

Example 7.9 in the Study Guide (pages 21-23 of Topic 7) shows the triangularisation procedure for a matrix. Consider the following matrix A, which also has eigenvalues 1, 1 and 5.

A =  l 」

「−1    −1     2

i. Construct a matrix S such that it is an orthogonal matrix with the first column corresponding

with the eigenvector  l 」0(1)

ii. Show that for the resulting matrix, S−1 AS = ], the 2 × 2 matrix A1  has eigenvalues

1 and 5 by determining and then simplifying its characteristic equation.

iii. Find the eigenvector from A1  corresponding with λ = 1 and then construct an orthogonal 2 × 2 matrix Q where the first column is based on your eigenvector. Hence construct the matrix

R =  l   0(1 0 0) 」

「   0      Q    .

iv. Calculate P = SR. Show that P is an orthogonal matrix and verify that P−1 AP is upper triangular.

5 marks

3    Section C - Choose 1 out of 3

Choose 1 of the following problems (if you complete more than 1, only the first submitted will be as- sessed). Write up your solution and results so that it could be understood by someone else completing SIT292, including all required working, investigations and relevant observations.  Make connections with the unit material where appropriate and provide references for any external resources used. High quality partial solutions and explorations can also receive high marks.

You will be given a mark out of 10 according to the following standards:

Mark    Description

9-10    Exceptional

7-8     Solution or partial solution is clearly presented with

sound understanding and insight demonstrated.

4-6     Solution demonstrates engagement with the problem,

good understanding and little in the way of misunderstandings

or reasoning errors.

2-3     Misunderstandings or errors in reasoning apparent in solution

provided. Some understanding demonstrated.

0-1     Not attempted or little to no understanding demonstrated.

1. From Theorem 7.1 in the study guide (and Corollary 1), we know that if we have n distinct eigenvalues for a n × n matrix, then we will be able to find linearly independent eigenvectors and hence diagonalise the matrix.

i. For a 2 × 2 matrix  [c(a)   d(b)], algebraically determine conditions that would lead to (a) distinct eigenvalues, (b) a repeated eigenvalue with only one non-linearly independent vector, and (c) a repeated eigenvalue with two linearly independent vectors.

ii. Hence provide original examples of each type.

iii. Can you find any similar conditions for the different cases for a 3 × 3 matrix?

10 marks

2. Use the R code below to generate and plot a random 2D dataset.  You can use R / R studio, or you can also copy then replace the code in one of the code windows in the online modules (a separate .txt file will be available in the assignment folder for this question so that you can

copy and paste it)

x  <-  rnorm(1000,0,2)    y  <-  rnorm(1000,0,2/3) data  <-  cbind(x,y)        ang  <-  pi/8

#  generate  normally  distr .  x-data, mean=0,  sd  =  2    #  generate  normally  distr .  y-data, mean=0,  sd  =  2/3 #  combine  x  and  y  data  together

#  set  an  angle  of  rotation

#  rotate  the  data

for(i  in  1:nrow(data))  {

data[i,]  <- matrix(

c(cos(ang),sin(ang),-sin(ang),cos(ang)),

nrow=2,ncol=2)  %*% matrix(data[i,],nrow=2)

}

plot(data)    #  generate  plot  of  the  data  after  rotation

cov(data)      #  calculate  the  covariance matrix

i. Take a screenshot of the plot and write down the values of the covariance matrix.

ii. Approximate the covariance matrix as  [1(3)   1(1)] and find the eigenvalues and eigenvectors.

iii. Add arrows to the plot representing your two eigenvectors.

vi. Adapt the code to make your own example with differently distributed data and follow steps i-iii.  Comment on any relationship(s) you observe between the covariance matrix values and the eigenvectors.

10 marks