Download Eigenvalues - Applied Linear Algebra - Exam Key and more Exams Linear Algebra in PDF only on Docsity!
SIMON FRASER UNIVERSITY
DEPARTMENT OF MATHEMATICS
Midterm 1
MATH 232 Fall 2011 Instructor: Prof. JF Williams Oct. 7 2011, 11:30 – 12:
Name: (please print) family name given name(s)
SFU IDs: student number @sfu.ca e-mail
Signature:
Instructions:
- Do not open this booklet until told to do so.
- Write your name above in block letters. Write your SFU student number and email ID on the line pro- vided for it.
- Write your answer in the space provided below the question. If additional space is needed then use the back of the previous page.
- 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.
- Calculators are not allowed. No books, papers, or electronic devices of any kind shall be within the reach of a student during the examination.
- During the examination, communicating with, or deliberately exposing written papers to the view of, other examinees is forbidden.
Question Points Score
Total 60
1. This problem concerns the matrices
A =
[
]
B =
[
]
C =
[
]
D =
[
]
[3] (a) What is AC + CD + DC?
[3] (b) Simplify and compute (B−^1 C)−^1 (B−^1 A)(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?
(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 x 1 , x 2 , x 3 and x 4.
[3] (c) Solve the system from part b).
[3] (d) Determine the maximum and minimum flow on each section.
3. In this problem you will construct a meal consisting of beans, cheese and rice.
- Beans contain 2 units carbohydrates, 2 units of protein and 1 unit of fat
- Cheese contains 1 unit of carbohydrates, 2 units of protein and 3 units of fat
- 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.
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 R^3 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.
