1. Discuss the different properties of Matrices for each matrix

2. What is the importance of interface in fiber reinforce composite?

The Oregon Atlantic company produces two types of paper: newsprint and wrapping paper. Make 1 yard of newspaper
It takes 5 minutes to produce one yard of newspaper; and 8 minutes to make one yard of wrapping paper. The company has 4,800 hours of operation per week. Newspapers cost $ 0.20 per yard and wrapping paper yields $ 0.25 per yard. Demand per week is 500 yards for newspaper and 400 yards for wrapping paper. sales price need labour hour is 1 hour.The company has established the following goals in order of importance:

(1) Limit overtime to 480 hours or less.

(2) Create a profit of $ 300 per week.

(3) profit size.

(4) Do not make fun of production capacity.

a. Determining how many different types of paper should be produced to satisfy various goals
Formulate it as a goal planning method model.

b. Use your computer to solve the problem.

1) How much will you have accumulated over a period of 30 years if, in an IRA which has a 10% interest rate compounded quarterly, you annually invest:

a. $1 b. $4000 c. $10,000 d. Part (a) is called the effective yield of an account. How could Part (a) be used to determine Parts (b) and (c)? (Your answer should be in complete sentences free of grammar, spelling, and punctuation mistakes.)

2) How much will you have accumulated, if you annually invest $1,500 into an IRA at 8% interest compounded monthly for: a. 5 year b. 20 years c. 40 years d. How long will it take to earn your first million dollars? Your answer should be exact rounded within 2 decimal places. Please use logarithms to solve.

Two balls are drawn in succession out of a box containing 3 red and 5 white balls. Find the probability that at least 1 ball was? red, given that the first ball was left parenthesis Upper A right parenthesis Replaced before the second draw. left parenthesis Upper B right parenthesis Not replaced before the second draw. ?(A) Find the probability that at least 1 ball was? red, given that the first ball was replaced before the second draw.

Find all ring homomorphism from Z⊕Z into Z⊕Z

A beverage company has introduced three new juice blends: carrot-canteloupe, kiwi-kale, and raspberry-rhubarb. They conducted a study where 210 volunteers tried all three blends and reported which ones they liked.

The number of volunteers who liked the carrot-canteloupe blend was 19.

The number of volunteers who liked the kiwi-kale blend was 18.

The number of volunteers who liked the raspberry-rhubarb blend was 21.

The number who liked both carrot-canteloupe and kiwi-kale was 14.

The number who liked both raspberry-rhubarb and kiwi-kale was 8.

The number who liked both carrot-canteloupe and raspberry-rhubarb was 10.

The number who liked all three blends was 6.

1. How many of the volunteers liked NONE of the three blends they tested?

2. How many liked just ONE of the blends?

3. How many liked kiwi-kale but not raspberry rhubarb?

A committe of four is being formed randomly of from the students and staff of a school: 5 administrators, 37 students and 4 teachers. How many ways can a committee be formed?

Suppose C is a m × n matrix and A is a n × m matrix. Assume CA = Im (Im is the m × m identity matrix). Consider the n × m system Ax = b.

1. Show that if this system is consistent then the solution is unique.

2. If C = [0 ?5 1

3 0 ?1]

and A = [2 ?3  

1 ?2   

6 10] ,,

find x (if it exists) when

(a) b =[1

0

3]

(b) b =[ 7

4

22] .

An investment of ​$71,000 was made by a business club. The investment was split into three parts and lasted for one year. The first part of the investment earned​ 8% interest, the second​ 6%, and the third​ 9%. Total interest from the investments was $5490.  The interest from the first investment was 6 times the interest from the second. Find the amounts of the three parts of the investment.

1.

f(x)= 3x / x^2 + 1

- Vertical Asymtote

As x →  ⁺, f(x) →

- Any hole in graph

- Horizontal asymtote

As x →  ⁺, f(x) →

Proof:

Let S ⊆ V be a subset of a vector space V over F. We have that S is linearly dependent if and only if there exist vectors v1, v2, . . . , vn ∈ S such that vi is a linear combination of v1, v2, . . . , vi−1, vi+1, . . . , vn for some 1 ≤ i ≤ n.

1----

. Is the statement​ "Elementary row operations on an augmented matrix never change the solution set of the associated linear​ system" true or​ false? Explain.

A.

​True, because elementary row operations are always applied to an augmented matrix after the solution has been found.

B.

​False, because the elementary row operations make a system inconsistent.

C.

​True, because the elementary row operations replace a system with an equivalent system.

D.

​False, because the elementary row operations augment the number of rows and columns of a matrix.

2---

Indicate whether the statements given in parts​ (a) through​ (d) are true or false and justify the answer.

a. Is the statement​ "Every elementary row operation is​ reversible" true or​ false? Explain.

A.

​False, because only interchanging is a reversible row operation.

B.

​True, because interchanging can be reversed by​ scaling, and scaling can be reversed by replacement.

C.

​False, because only scaling and interchanging are reversible row operations.

D.

​True, because​ replacement, interchanging, and scaling are all reversible.

3----

In parts ​(a) through​ (e) below, mark the statement True or False. Justify each answer.

a. The echelon form of a matrix is unique. Choose the correct answer below.

A.

The statement is true. The echelon form of a matrix is always​ unique, but the reduced echelon form of a matrix might not be unique.

B.

The statement is true. Neither the echelon form nor the reduced echelon form of a matrix are unique. They depend on the row operations performed.

C.

The statement is false. The echelon form of a matrix is not​ unique, but the reduced echelon form is unique.

D.

The statement is false. Both the echelon form and the reduced echelon form of a matrix are unique. They are the same regardless of the chosen row operations.

(a) The n × n matrices A, B, C, and X satisfy the equation AX(B + CX) ?1 = C Write an expression for the matrix X in terms of A, B, and C. You may assume invertibility of any matrix when necessary.

(b) Suppose D is a 3 × 5 matrix, E is a 5 × c matrix, and F is a 4 × d matrix. Find the values of c and d for which the statement “det(DEF) = 1” can be valid. Explain your answer.

(c) Find all (real or complex) values of x such that the matrix GH is invertible, where G =

x^2, ?1

x , x ? 2

H = x ? 1 , ?2

1 , x + 1

An investment club has set a goal of earning 15% on the money they invest in stocks. The members are considering purchasing three possible stocks, with their cost per share (in dollars) and their projected growth per share (in dollars) summarized in the table. (Let x = computer shares, y = utility shares, and z = retail shares.)
Stocks
Computer (x)   Utility (y)   Retail (z)
Cost/share   30   44   26
Growth/share   6.00   6.00   2.40
(a) If they have $392,000 to invest, how many shares of each stock should they buy to meet their goal? (If there are infinitely many solutions, express your answers in terms of z as in Example 3.)

b) If they buy 1500 shares of retail stock, how many shares of the other stocks do they buy?
computer

c)What if they buy 3000 shares of retail stock?
computer
utility

(c) What is the minimum number of shares of computer stock they will buy?

D)What is the number of shares of the other stocks in this case?
utility
retail

(d) What is the maximum number of shares of computer stock purchased?

E)What is the number of shares of the other stocks in this case?
utility
retail