Reference no: EM13843334
1. Use Gaussian elimination to determine if each of the following linear systems is consistent. Then use Gauss-Jordan elimination to solve the consistent linear systems. Use Notion 1.4.9 to denote the elementary row operations that you used for each step.
a. 2x - 2y + 4z = 5
x + 3y + 6z = 2
x + 7y + 10z = 1.
b. x - 2y + z = 4
2x + 6y + 3z = 5
-x + 2y - 7z = 2.
c. x1 + 3x2 - x3 + 4x4 - x5 = 2
2x1 + x2 + 4x3 - 3x4 + 5x5 = 1
-x1 + 2x2 + 3x3 - x4 + 2x5 = 9.
2. Determine the values of λ so that the following homogenous linear system has only the trivial solution:
λx + y + z = 0
x + λy + z = 0
x + y + λz = 0.
3. For which real numbers a and b does the following linear system have (a) no solution, (b) exactly one solution, or (c) infinitely many solutions? Justify your answers.
(a - 1)x + (a + 3)y + z = 1
ax + a2y + z = 1
x + ay + z = b.
4. A certain library owns 10 000 books. Each month 20% of the books in the library are lent out and 80% of the books lent out are returned, while 10% remain lent out and 10% are reported lost. Finally, 25% of the books listed as lost the previous month are found and returned to the library. At present all the books are in the library. How many books will be in the library, lent out, and lost after three months?
5. Consider an economy with three industries: coal-mining operation, oil-drilling operation and electricity-generating plant.
i) To produce $1.00 of coal, the mining operation must purchase $0.10 of its own production, $0.40 of oil and $0.10 of electricity.
ii) To produce $1.00 of oil, it takes $0.20 of coal, $0.20 of oil and $0.20 of electricity.
iii) To produce $1.00 electricity, the electricity-generating plant must purchase $0.20 of coal, $0.40 of oil and consume $0.20 of electricity.
Assume that during one week, the economy has an exterior demand of $50,000 worth of coal, $75,000 worth of oil, and $100,000 worth of electricity. Find the production of each of the three industries in that week to exactly satisfy both the internal and the external demands.
6. For any real number θ, define the matrix Aθ =
(cos θ - sin θ)
(sin θ cos θ)
(i) Prove that Aθ1Aθ2 = Aθ1+θ2 .
(ii) Let B = ( 1 0 0 -1) . Prove that BAθ = A-θB.
7. Let A and B be diagonal matrices of order n whose diagonal entries are
a1, a2, . . . , an and b1, b2, . . . , bn,
respectively. Prove that AB is the diagonal matrix whose diagonal entries are
a1b1, a2b2, . . . , anbn.
[Hint: Let A = (aij )n×n. Then aij = ai if i = j, and aij = 0 if i 6= j.]
8. Let A = (aij )n×n such that AB = BA for every square matrix B of order n.
(i) Prove that A is a diagonal matrix.
[Hint: Let Bi denote the matrix whose (i, i)-entry is 1 and 0 elsewhere. Show that aij = 0 whenever i 6= j.]
(ii) Prove that A is a scalar matrix.
[Hint: For i 6= j, let Bij denote the matrix whose (i, j)-entry is 1 and 0 elsewhere. Show that aii = ajj for all i 6= j.]
|
Exponential smoothing forecasting function
: If we forecast $100,000 sales for July, we actually sold $200,000 in July, and we want an exponential smoothing forecasting function with the highest noise dampening among @=0.1, @=0.3, @=0.5; if you can, forecast August sales. If you cannot, state w..
|
|
Decision making occur in the business environment
: Problems and decision making occur in the business environment on a daily basis. Think about the street vendor in New York City on up to the CEO of a Fortune 500 company. Each faces many problems and each has clients that he or she has to work and in..
|
|
Computer security is a branch of technology
: Computer security is a branch of technology known as information security as applied to computers. The objective of computer security varies and can include protection of information from theft or corruption, or the preservation of availability, as d..
|
|
Summarize the reasons that employees resist change
: Summarize the reasons that employees resist change. How would you as a leader prevent or deal with this employee resistance?
|
|
Linear systems is consistent
: 1. Use Gaussian elimination to determine if each of the following linear systems is consistent. Then use Gauss-Jordan elimination to solve the consistent linear systems. Use Notion 1.4.9 to denote the elementary row operations that you used for ea..
|
|
Describe lewins model of change
: Describe Lewin’s model of change. Can you use this model to relate to a change you have had to make in your own life?
|
|
Primary functions of our health care system
: Diagnosis and treatment were the primary functions of our health care system. Why was that the case? What has caused that to change?
|
|
Strategies of corporate procurement and project procurement
: Compare and contrast the two (2) basic procurement strategies of corporate procurement and project procurement. Illustrate the risk(s) associated with using corporate procurement and project procurement.
|
|
Describe three components of the patients bill of rights
: Describe three components of the Patient’s Bill of Rights. Have you experienced or observed a violation of any of these? Provide a brief summary.
|