Question 13 (1 point)

Every invertible matrix is diagonalizable.

Question 13 options: True False

Question 14 (1 point)

Every diagonalizable matrix is invertible.

Question 14 options: True False

Let Q1=y(1.1), Q2=y(1.2), Q3=y(1.3), where y=y(x) solves...

1) y'''+2y''-5y'- 6y=4x^2 where y(0)=1, y'(0)=2, y''(0)=3

2) y'''- 6y''+11y'- 6y=6e^(4x) where y(0)=4, y'(0)=10, y''(0)=30

3) y''- 6y'+9y=4e^(3x) ln(x) where y(1)=, y'(1)=2

Please show all steps and thank you!!!

solve for matrix B

Let I be Identity matrix

(I-2B)^-1=

Discrete math

Summarize all the theorem regarding to graph theory.

e.g.) A connected graph has a Euler circuit iif every vertex is of even degree.

Let R be the relation on Z+× Z+ such that (a, b) R (c, d) if and only if ad=bc. (a) Show that R is an equivalence relation. (b) What is the equivalence class of (1,2)? List out at least five elements of the equivalence class. (c) Give an interpretation of the equivalence classes for R. [Here, an interpretation is a description of the equivalence classes that is more meaningful than a mere repetition of the definition of R. Hint: Look at the ratio a/b corresponding to (a, b).

Can ?⃑⃑ = ( 11 −5 9 ) be written as a linear combination of the vectors ?⃑ 1⃑ = ( 1 0 1 ), ?⃑⃑2⃑ = ( −2 3 −2 ), and ?⃑⃑3⃑ = ( −6 7 5 )? Does your answer allow you to conclude whether or not ? = {?⃑ 1⃑, ?⃑⃑2⃑, ?⃑⃑3⃑} is a basis for ? 3 ?

Create a vector ?⃑ such that the set {( 1 0 0 ) , ( 1 3 2 ) , ?⃑} is a basis for ? 3 . Explain.

ANSWER BOTH PARTS WILL RATE

Prove the following statements by using the definition of convergence for sequences of
real numbers.

a) If {cn} is a sequence of real numbers and {cn} converges to 1 then {1/(cn+1)} converges to 1/2

b) If {an} and {bn} are sequences of real numbers and {an} converges A and {bn} converges to B and B is not equal to 0 then {an/bn} converges to A/B

*NUMBER THEORY*

1.Find all the possible solutions for the following diphantine equations by using the euclidian algorithim. You must show all the process to get credit.

a.           3x + 5y = 7

b.           3x − 12y = 7

c.           1990x − 173y = 11

d.           21x + 48y = 6

e.           2x + 3y + 5z = 11

Winkler Furniture manufactures two different types of china cabinets: a French provincial model and a Danish Modern model. Each cabinet produced must go through three departments: carpentry, painting, and finishing. The table below contains all relevant information concerning production times per cabinet produced and production capacities for each operation per day, along with net revenue per unit produced. The firm has a contract with an Indiana distributor to produce a minimum of 300 of each cabinet per week (or 60 cabinets per day). Owner Bob Winkler would like to determine a product mix to maximize his daily revenue.

Formulate as an LP problem and obtain the revenue      

what is the principal on a $10,000 loan at a rate of 7% number of compounding per year is 12 term in years is 18 months what is the payment and the total pay of interest paid
how do you find the monthly payment.

1. A consumer electronics company is planning to introduce a new device. After careful consideration of costs, (e.g., there is a fixed cost of $5 million for developing the item),  the projected state of the economy, etc., the marketing manager came up with the following payoff table (in $millions)

1. Calculate the regret table below. SHOW ALL WORK TO GET FULL CREDIT. (6 pts.)

2. Calculate the minimax regret below. SHOW ALL WORK TO GET FULL CREDIT. (3 pts.)

1. Using the following table of relation between X & Y (Y is the independent Variable):

xi -2 2 4 5 4 6 7 8 11

yi -4 1 3 1 3 8 10 20 30

a. Find the estimated regression equation. Show ALL your calculations and the appropriate formula. WRITE Formula in all the questions below and substitute for the numbers, demonstrate your work.

b. Calculate R2. Comment on the Goodness of the Fit. Is it a good fit? Why?

c. Test the hypothesis regarding the significance of the slope β1. (Use α = 0.05) Show ALL the calculations necessary to get your answer. Hint: You need to calculate t by using all the formulas needed for it using t in the Formula Sheet.

d. Calculate the values of F, using formula for F statistics. e. Draw the scatter diagram and also your estimated regression line in a diagram.