Row Equivalent Matrices and Systems of Equations

Determine the solutions of the system of equations whose matrix is row-equivalent to: Give three examples of the solutions. Verify that your solutions satisfy the original system of equations. Show work.

Solving Systems of Equations with Three Variables

Solve the following systems of equations: x1 + x2 = 1 -x1 + x2 + x3 = -1 -1x2 + x3 = 3 -Show all necessary work.

Matrix Inverses

Show that: (Matrix) 2 3 -1 1 2 1 -1 –1 3 is the inverse of 7 -8 5 -4 5 3 1 -1 1 Show all work.

Proof by Induction : Step-by-step

Let p(n) be the statement that: 1^3 + 2^3 + ... + n^3 = (n (n + 1) /2)^2 for the positive integer n. a) What is the statement P(1)? b) Show that P(1) is true, completing the basis step of the proof. c) What is the inductive hypothesis? d) What do you need to prove in the inductive step? e) Complete the inductive step. f) Explain why these steps show that this formula is true whenever n is a positive integer. Show all work. See attached file for full problem description.

Probability Game of Shooting

In a single duel, Shooter A hits target with a probability of 1/3. B with 1/2. C never misses. ABC fire in turns, A going first, B next, and C last. Shooting cycle repeats until only one dueler has not been hit. Use random # x, where 0 <= x <=1. What happens if the shooter hits the deadliest shooter first? in 10,000 duels, how many will A win? B win? C ? What happens if the first shot is intentionally missed? What is the better strategy shooting the deadliest one or the weakest one.? Who has better chance of winning the worst or the best shooter?