MA 225 Boston University Linear Algebra equations

Anonymous
timer Asked: Feb 18th, 2019
account_balance_wallet $20

Question Description

SHOW ALL OF YOUR WORK


good luck

Name: February 17, 2019 MA 225-01 Spring 2019 Linear Algebra Take Home Exam #1 – Questions will be posted on Blackboard by Sunday, February 17th. I will send an announcement via Blackboard (to your udc.edu email) once posted. SHOW ALL OF YOUR WORK (use additional sheets of paper if necessary) The exam will be due at 11:00am in class on Tuesday, February 26, 2019. ATTACH THIS COVER SHEET TO YOUR EXAM ANSWERS Please fill in your name and sign your name below to notify Dr. Barnette and UDC that you followed this mandatory requirement: I, _____________________________, attest and confirm that I did not collaborate with anyone on this Take Home Exam. I only used my MA 225 textbook references, presentation handouts posted on Blackboard, or class notes/assignments to complete this MA 225 Take Home Exam. __________________________________________ Signature _________________ Date (please see following pages 2 through 5 for exam questions; there are 3 exam questions; each question has multiple parts) SHOW ALL OF YOUR WORK (use additional sheets of paper if necessary) 1 Dr. Kim Barnette MA 225 Spring 2019 1. (worth 10 points) A system of linear equations only has three possibilities of a solution set. a. What are those three possibilities? b. Cite the theorem from section 1.6 that states this fact. 2. (worth 40 points) Consider the system of linear equations below of the form Ax =b x1 + 2x2 + 3x3 = 5 2x1 + 9x2 + 3x3 = 12 x1 + + 4x3 = 7 a. Write out the coefficient matrix A. b. Write the column vector b. c. Determine if A is nonsingular by finding the determinant of A. Use either arrow technique or co-factor expansion along any row or column of A of your choice. d. Is A invertible and why or why not? 2 Dr. Kim Barnette MA 225 Spring 2019 e. Solve the system of linear equations above by inverting the coefficient matrix and using Theorem that states: If A is an invertible n x n matrix, then for each n x 1 matrix b, the system of equations Ax = b has exactly one solution, namely, x = A-1b. i. Find the inverse of A by applying to the identity matrix an appropriate set of row operations. ii. Find the inverse of A by applying the determinant and classical adjoint method. f. Find the solution x = A-1b and show/check that the solution x solves the system. 3 Dr. Kim Barnette MA 225 Spring 2019 g. Show how to use Cramer’s rule to solve the linear system above. Do you get the same values for x1, x2 and x3 as you did in part (f)? 3. (worth 50 points) A corporation wants to lease a fleet of 12 airplanes with a combined carrying capacity of 220 passengers. The three available types of planes carry 10, 15, and 20 passengers, respectively. Follow the steps below to find out how many of each type of plane should be leased. a. Let x1 = number of 10-passenger planes; Let x2 = number of 15-passenger planes; Let x3 = number of 20-passenger planes. What are the two linear equations that you must solve? b. Write the augmented coefficient matrix from the system of linear equations in part a. c. Solve the system of linear equations by Gauss-Jordan elimination. 4 Dr. Kim Barnette MA 225 Spring 2019 d. You should result in a solution where the solution to x1 and x2 depends on x3. Let x3 = t, where t is any real number. Since you cannot lease fractional planes or negative planes, find the possible values for t. e. Given the possible values of t found in part (d), create a table to list the possible solutions for x1, x2, x3. f. If the cost of leasing a 10-passenger plane is $8,000 per month, a 15-passenger plane is $14,000 per month, and a 20-passenger plane is $16,000 per month, which of the three possible solutions in your table from part (3e) would minimize the monthly leasing cost? 5 Dr. Kim Barnette MA 225 Spring 2019

Tutor Answer

Bilal_Mursaleen
School: Purdue University

Here's the solution file.:) :)

1. A system of linear equations only has three possibilities of a solution set.
a. What are those three possibilities?
Ans: - There are three types of solutions of the system of linear equations.
• No solution
• Unique solution (one solution)
• Infinitely many solutions
b. Cite the theorem from section 1.6 that states this fact.
“The theorem will be given in section 1.6 of your book”
2. Consider the system of linear equations below of the form Ax =b
𝑥1 + 2𝑥2 + 3𝑥3 = 5
2𝑥1 + 9𝑥2 + 3𝑥3 = 12
𝑥1 + 4𝑥3 = 7
a. Write out the coefficient matrix A.
Sol: 1 2 3
𝐴 = [ 2 9 3]
1 0 4
b. Write the column vector b.
Sol: 5
𝑏 = [12]
7
c. Determine if A is nonsingular by finding the determinant of A. Use either
arrow technique or co-factor expansion along any row or column of A of your
choice.
Sol: Using co-factor expansion along first row to find determinant, we get

1 2 3
|𝐴| = |2 9 3|
1 0 4
9 3
2 9
2 3
|𝐴| = (−1)1+1 (1) |
| + (−1)1+2 (2) |
| + (−1)1+3 (3) |
|
0 4
1 4
1 0
9 3
2 9
2 3
|𝐴| = (−1)2 (1) |
| + (−1)3 (2) |
| + (−1)4 (3) |
|
0 4
1 4
1 0
9 3
2 9
2 3
|𝐴| = 1 |
| − 2|
| + 3|
|
0 4
1 4
1 0
|𝐴| = 1[(9)(4) − (0)(3)] − 2[(2)(4) − (1)(3)] + 3[(2)(0) − (1)(9)]
|𝐴| = 1[36 − 0] − 2[8 − 3] + 3[0 − 9]
|𝐴| = 1[36] − 2[5] + 3[−9]
|𝐴| = 36 − 10 − 27
|𝐴| = −1
Hence, matrix A is non-singular.
d. Is A invertible and why or why not?
Ans: - A matrix is said to be invertible if and only if it has non-zero determinant.
So, according to definition, matrix A is invertible because its determinant is
non-zero.
e. Solve the system of linear equations above by inverting the coefficie...

flag Report DMCA
Review

Anonymous
Goes above and beyond expectations !

Similar Questions
Hot Questions
Related Tags
Study Guides

Brown University





1271 Tutors

California Institute of Technology




2131 Tutors

Carnegie Mellon University




982 Tutors

Columbia University





1256 Tutors

Dartmouth University





2113 Tutors

Emory University





2279 Tutors

Harvard University





599 Tutors

Massachusetts Institute of Technology



2319 Tutors

New York University





1645 Tutors

Notre Dam University





1911 Tutors

Oklahoma University





2122 Tutors

Pennsylvania State University





932 Tutors

Princeton University





1211 Tutors

Stanford University





983 Tutors

University of California





1282 Tutors

Oxford University





123 Tutors

Yale University





2325 Tutors