For matrices, a mulitplicative identity is a square matrix X such XA = AX = A for any square matrix A. Prove that X must be the identity matrix.

Prove that for any invertible matrix A, the inverse matrix must be unique. Hint: Assume that there are two inverses and then show that they much in fact be the same matrix.

Prove Theorem which shows that Gauss-Jordan Elimination produces the inverse matrix for any invertible matrix A. Your proof cannot use elementary matrices (like the book’s proof does).

Prove that null(A) is a vector space.

Prove that col(A) is a vector space.

Linear Algebra

we know that x ∈ R^n is a nonzero vector and C is a real number.

find all values of C such that ( In − Cxx^T ) is nonsingular and find its inverse

knowing that its inverse is of the same form

Section 5.3: Find the equilibrium values of a general quadratic population model:

dx/dt=a1x+b1x^2+c1xy

dy/dt=a2y+b2y^2+c2xy

Don’t forget to show this for all four cases:

• Case 1: ? = 0, ? = 0
• Case 2: ? ≠ 0, ? = 0
• Case 3: ? = 0, ? ≠ 0
• Case 4: ? ≠ 0, ? ≠ 0 (Use Cramer’s Rule to set up the solution for this case.)

(A) Find the general solution for the displacement x = x(t) of the forced mechanical system x´´ + 6x´ + 8x = 35 sin t. (B) Identify the steady-periodic part

Suppose G is a connected cubic graph (regular of degree 3) and e is an edge such that G − e has two connected components G1 and G2

(a) Explain what connected means.

(b) We say that e is a____________ of G

(c) show that G1 has an odd number of vertices.

(d) draw a connected cubic graph G with an edge e as above.

y''+xy'+(2x^2+1)y=0

Use power series to find at least four terms of the following differential equations

. Assume that Randall Ezno was born today and to celebrate his birth, his parents deposited $10,000 into an account in his name. The account pays interest of 2.35 percent p.a., with monthly compounding, and it is expected to continue paying this amount forever. Assume that exactly 1.5 years after the deposit was made, Randall’s parents changed their minds and withdrew $10,000 from the account (all that remained in the account was the interest earned in the first 18 months). How much money will be in Randall’s account on his 65th birthday?

Perform four iterations, if possible, on each of the functions g defined in Exercise 1. Let Po=1 and P(n+1)=g(Pn), for n=0,1,2,3.

b. Which function do you think gives the best approximation to the solution?

heres the functions g defined in exercise 1.

g1(x)=(3+x-2x^2)^1/4

g2(x)=(x+3-x^4/2)^1/2

g3(x)=(x+3/x^2+2)^1/2

g4(x)=3x^4+2x^2+3/4x^3+4x-1

Let E be the solid region below the sphere x^2 + y^2 + z^2 = 16, above the cone z = Sqrt[x^2 + y^2], and by the planes x = 0, y = 0, and z = 0 in the first octant.

Compute the Triple Integral [(x+y+z)Cos(x^2+y^2+z^2)]dV on the region E.

Please set up the integral.

Ex 1. Prove by contrapositive the following claims (please, write down the contrapositive for each statement first).

Claim 1: Let n be an integer. If n 2 − 6n + 5 is even, then n is odd.

Claim 2: Let a, b, c be positive real numbers. If ab = c then a ≤ √ c or b ≤ √ c.

Prove that it is possible to make up any postage of n-cents using only 5-cent and 9-cent stamps for n ≥ 35.

Sasha, a doctor based in Toronto had to stop working temporarily due to the health and safety restrictions imposed by the government due to the outbreak of the swine flu. She started to make two types of children's wooden toys in her basement, cars (C) and dolls (D). Cars can yield a contribution margin of $9 each and dolls have a contribution margin of $8 each. Since her electric saw overheats, Sasha can make no more than 7 cars or 14 dolls each day. Since she doesn't have equipment for drying the lacquer finish that she puts on the toys, the drying operation limits her to 16 cars or 8 dolls per day. Sasha wants to know what is the combination of cars and dolls that she should make each day to maximize her profits, subject to her constraints. a) Formulate a linear programming problem to depict Sasha's objective algebraically. b) What are the corner points in Sasha's feasible region? [Note: You do not need to graph the feasible region in your answer file, but you may want to draw it in a scrap paper to understand the problem better and solve it c) Solve this problem using the corner point method. Find the optimal combination of cars and dolls. What is the optimal profit at that combination?

Marriott Hotel data breach, explain with a report on it that answers the following question; When and where was the breach? What was the cause of the breach (be specific)? Who was affected? How severe were the effects of the data breach on the individuals affected? Were there other potential effects (e.g. identity theft) that haven't been documented yet? How did the company respond to the breach? Was there anything the company failed to do that would have prevented it? Did the company undertake any actions to prevent this (or similar things) from happening again? Was there a public outcry over the company's response to the breach? How did the company address the public's reaction? What consequences did the company or individuals at the company suffer (firings, legal liabilities, etc.)? What recommendations would you make to prevent a similar breach from occurring?.

The weight of apples is normally distributed. Large apples have a mean of 15 oz and medium apples have a mean of 10 oz. The standard deviation of the weight of both large and medium apples is 2 oz.

You select a large apple and a medium apple at random. Let L be the weight of the large apple and M be the weight of a medium apple.

Let X be the total weight of the two apples (X = L + M). The distribution of X is also normal.

1. Use the rules for means and variances to find the mean and standard deviation of X.

2. What is the probability that the total weight X is between 22 and 28 oz?

we have the question that 'if F is a field with char 0, prove that prime subfield of F is isomorphic to the field of Q'. I already figure out the answer. BUT from the question I have other question have risen in my brain. 1. what are the official definition of the kernel of a map and the characteristics of a field? 2. what is the link between the kernel and the char? 3. are they equivalent in some context?