M230

Test III

Fall 1997

Bellamy

Point values are as indicated.

1. 20 Points

   Given that the inverse of is , solve both of the following systems by doing the appropriate matrix multiplications. Then answer the question in part c.
   
   1. 5x + y = 156
      8x + 12y = 468
      
      
   2. 5x + y = 156
      8x + 12y = 520
      
      
   3. Which system solution, a or b, had the larger value of x?
   
   
   

2. 20 Points

   Find the inverse of the following matrix showing all your row operations:
   
   

3. 30 Points

   You know that x 0; y 0; x + 2y 40; y 2x ; x 3y. Find graphically the maximum possible value of y, the maximum possible value of x, and the maximum possible value of x + y. Which two of these occur at the same corner point?
   
   

4. 20 Points

   An urn contains 3 balls; one red and two green to start with. A ball is drawn randomly and its color noted. Then a ball of the opposite color is put back into the urn in place of the one drawn, so that the urn always contains a total of three balls. Letting the state of this system be denoted by the number of red balls in the urn at any step, how many states are there? (The system starts in state #1, that is, with one red ball.)
   
   
   1. Draw a transition diagram for this Markov chain.
   2. What is the probability that the system is in state #1 after one draw?
   3. What is the probability that the system is in state #1 after exactly two draws?

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