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SIMON FRASER UNIVERSITY

DEPARTMENT OF MATHEMATICS

Midterm 1

MATH 232 Fall 2011

Instructor: Prof. JF Williams

Oct. 7 2011, 11:30 – 12:20

Name: (please print) family name given name(s)

SFU IDs: student number @sfu.ca e-mail

Signature:

Instructions:

1. Do not open this booklet until told to do so.

2. Write your name above in block letters. Write your SFU student number and email ID on the line pro- vided for it.

3. Write your answer in the space provided below the question . If additional space is needed then use the back of the previous page.

4. This exam has 5 questions on 6 pages (not includ- ing this cover page). Once the exam begins please check to make sure your exam is complete.

5. Calculators are not allowed. No books, papers, or electronic devices of any kind shall be within the reach of a student during the examination.

6. During the examination, communicating with, or deliberately exposing written papers to the view of, other examinees is forbidden.

Question Points Score

1 12

2 12

3 12

4 12

5 12

Total 60

MATH 232 Page 1 of 6

1. This problem concerns the matrices

A =

[ 2 −1 3 0

] B =

[ 2 −1 1 0

] C =

[ 1 −1 1 0

] D =

[ 2 −1 −4 2

] [3] (a) What is AC + CD +DC?

[3] (b) Simplify and compute (B−1C)−1(B−1A)(BA)−1

[3] (c) Using your result from part b) explain why the system BCx = b has a unique solution for all b.

[3] (d) For what b does Dx = b have infinitely many solutions?

MATH 232 Page 2 of 6

Figure 1: The arrows indicate the direction of one way streets and cannot be reversed.

2. This question relates to the figure above which describes traffic flow on some downtown streets.

[3] (a) By considering only the total flow in and out determine the flow (both magnitude and direction) through the traffic light at point y.

MATH 232 Page 3 of 6

(Question 2 continued)

[3] (b) By balancing each of the four nodes, write down a system of four equations of the form Ax = b to determine the flow through the points x1, x2, x3 and x4.

[3] (c) Solve the system from part b).

[3] (d) Determine the maximum and minimum flow on each section.

MATH 232 Page 4 of 6

3. In this problem you will construct a meal consisting of beans, cheese and rice.

1. Beans contain 2 units carbohydrates, 2 units of protein and 1 unit of fat

2. Cheese contains 1 unit of carbohydrates, 2 units of protein and 3 units of fat

3. Rice contains 3 units of carbohydrates, 1 unit of protein and 0 units of fat

[4] (a) Set up a system of three linear equations whose solution x tells the quantities of beans, cheese and rice needed to provide 8 units of carbohydrates, 7 units protein and 7 units of fat. (Be sure to indicate what the entries of your vectors correspond to!)

[2] (b) Write your linear system in the form Ax = b

[6] (c) Solve matrix system above to determine how many units of beans, cheese and rice are needed to meet the target.

MATH 232 Page 5 of 6

4. This question concerns the homogeneous solutions to the system with coefficient matrix

A =

−1 0 1 0 2 1 0 1 0 1 2 1 2 2 2 2

[6] (a) Write the solution x to Ax = 0 in the form x = sv1 + tv2 with scalars s, t.

[2] (b) What geometric object is this solution space?

[4] (c) Use only the matrix A to determine if the vector u = [1, 2,−1,−2]T is in span{v1,v2}. (Note that u is a column vector!)

MATH 232 Page 6 of 6

5. Short answer questions, put your answer in the box. An answer of just True or False with no explanation will receive no credit.

[3] (a) True or False: Two nonparallel lines in R3 must intersect at at least one point.

[3] (b) True or False: A linear system with three equations and two unknowns may have a unique solution.

[3] (c) True or False: If both matrix products AB and BA are defined then A and B are both square matrices.

[3] (d) True or False: If Ax = b is inconsistent then Ax = 0 only has the zero solution.