代做Math 112 Assignment 4代写Processing

Math 112

Assignment 4

Due 3 April 2024

Submit your answers to the following questions in the appropriate place in Crowdmark.

1. (a) Consider the matrix  Find a formula for An , for n = 1, 2, . . ..

(b) The sequence (a0, a1, . . . , an, . . .) is defined as follows:

a0 = 2, a1 = 1 and an+1 = 5an − 6an−1 for all n ≥ 0.

Find the expression for an (in terms of n ).

2. (a) Find a matrix P that diagonalizes the given matrix 

(b) Determine the solution to the dynamical system defined as follows::

with the initial condition

Write separate equations for xn, yn, and zn.

3. Consider the “win or lose” game with the following transition matrix.

(a) Let pK be the probability of eventually winning from one of the nodes K. Find pK for K = X, Y by solving a two equations which relates the pK’s.

(b) Again by solving two equations, calculate the average number of moves tK required to end the game starting on node K for K = X, Y .

(c) Find the eigenvalues of the game matrix A, and a set of 4 linearly independent eigenvectors. Use these to find a formula for the probability of being in any of the four states after n moves starting on node Y . [Hint: 1 is a repeated eigenvalue for the matrix A.]

(d) Use your formula from part (c) to compute pY by an alternate method. (Explain the method.) Similarly, compute tY by using your formula from part (c).

4. A and B play a game with the following rules: A starts with $1, and B starts with $2. A fair die is rolled to determine the outcome of each turn:

• If a 1, 2, or 3 is rolled, B gives A $1.

• If a 4 is rolled, B gives A all its money.

• For any other outcome, A gives B $1.

The game ends when a player runs out of money, and this player loses the game.

Consider the following states in your analysis:

• X: A has $1

• Y: A has $2

• W: A has $3

• L: A has $0

(a) Find the transition matrix of the given states.

(b) Calculate the probability of A winning the game.

(c) Determine the expected length of the game.