ESI6314: Deterministic Methods in Operations ResearchHW #1
Please solve the problems by hand, instead of using computers/calculators, except the case in which the
question explicitly requires it. You do not need to type the answers. Handwriting should be sufficient.
Question 1
Given the matrix
1
A= 0
−1
0
−1
0
1
0 ,
1
compute
1. B1 = (−A)T ,
2. B2 = A + (−A)T ,
3. B3 = (−A)T × A,
4. B4 = A × (−A)T .
Question 2
If 120 pounds of aluminum can be used to make 3 mountain bikes and 4 racing bikes, and 150 pounds
of aluminum can be used to create 5 mountain bikes and 3 racing bikes; determine how many pounds of
aluminum are used to make each mountain bike and how many are used for each racing bike.
Question 3
Consider the following system of linear equalities
x1
b
3 0 −1
1 a 0 x2 = 1
1
0 2 1
x3
where a and b are given real numbers. Using Gauss-Jordan elimination, determine conditions on a and b
under which
1. The system has no solution.
2. The system has an infinite number of solutions.
3. The system has a unique solution.
When the system is not infeasible, describe all of its solutions.
1
Question 4
Consider the system of equations
ax1 + 3×2
=
6a
x1 + 2×2
=
6.
For the values a = 0, a = 1, and a = 2, represent in the plane the lines corresponding to these equations,
and highlight on the picture the solution of the corresponding system. Determine whether there exists a
value of a for which the system is infeasible.
Question 5
Consider three vectors v1 = [1, 1, 1], v2 = [a, 1, b], and v3 = [1, a + b, 0], where a and b are parameters. Find
all conditions on the values of a and b (if any) for which:
1. the number of linearly independent vectors in this collection is 1.
2. the number of linearly independent vectors in this collection is 2.
3. the number of linearly independent vectors in this collection is 3.
Question 6
If
3
A= 6
9
0
0
0
x1
1
4
2 , x = x2 , and b = 8 ,
3
x3
12
how many free variables do you have in a solution set when you solve the linear system Ax = b?
Question 7
Find the determinant of the following square matrix
1
0
A=
0
1
0 1
2 0
0 −1
0 0
0
0
.
0
1
Question 8
Consider the matrix and vector
1
A= 1
0
0
1
1
−1
−1
a and d = 0 ,
1
1
where a 6= 0. Use Gauss-Jordan elimination to compute A−1 . Use this result to solve the system Ax = d.
Also, use the excel sheet to obtain A−1 for a = 1.
2
Question 9
Consider the system Ax = b where matrix
1
0
A=
0
0
0
A and vector b are defined as
1
2 3 4 5
2
6 7 8 7
0 6 5 4 and b =
0
0
0 0 3 2
3
0 0 0 1
Show that this system has a single solution (Call it x∗ ). Then computer the value of x∗3 .
Question 10
Let a jar contain a quantity of nickels, dimes and quarters. There are a total of 30 coins in the jar. The
totally value of the coins is 3 dollars. There are three times as many nickels as quarters. How many coins
of each type are in the jar?
Remark: If you feel there is some confusion concerning any question, please write down your understanding
of the problem and write the corresponding answer.
3
Delivering a high-quality product at a reasonable price is not enough anymore.
That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.
You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.
Read moreEach paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.
Read moreThanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.
Read moreYour email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.
Read moreBy sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.
Read more