Problem 10 Explain and answer Ch 9. Write the rabbit-fox equations in matrix for

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Problem 10
Explain and answer

Ch 9. Write the rabbit-fox equations in matrix form using PR, Fl, p'R FT and 15 10. Consider the system of equations x1 + 4x2 + 3x3 6i-4y2 + 4y3-120 (a) Write this system of equations in matrix form. Define the vectors and matrices you introduce. (b) Rewrite in matrix form with all the variables on the left side (and just numbers on the right) 11·Three different types of computers need varying amounts of four dif- ferent types of integrated circuits. The following matrix A gives the number of each circuits needed by each computer. Circuits 1 2 3 4 A 2 3 2 1 A= Computers B15 1 3 2 cL3 222] Let d [10, 20, 30] be the computer demand vector (how many of each type of computer is needed). Let p = [$2, $5, $1, $101 be the price vector for the circuits (the cost of each type of circuit). Write an expression in terms of A, d, p for the total cost of the circuits needed to produce the set of computers in demand; indicae where the matrix-vector product occurs and where the vector product occurs. Compute this total cost. 1.2 12. For the frog Markov chain in Example 6 it was noted in Section I rkov tran that p* = [.1, .2, .2, .2, .2, .1] is a stable dio,ribution. In matrix algebra, this means that p Ap*, where A sition matrix. Verify that p* = Ap* for th ector by 13. One can express polynomial product as follows: to mul x2x 5, we multiply tion uadratic

Explanation / Answer

Solution:

Splitting both sides into vector/matrix form we get

2x1 + 3x2 2x3 = 5y1 + 2y2 3y3 + 200

x1 + 4x2 + 3x3 = 6y1 4y2 + 4y3 120

5x1 + 2x2 x3 =2y1 + 0y2 2y3 + 350

we can then split the right-hand vector into a sum of a vector with vector c = [200, 120, 350] . yi terms (i=1,2,3) and a constant

2x1 + 3x2 2x3= 5y1 + 2y2 3y3

x1 + 4x2 + 3x3 = 6y1 4y2 + 4y3 + c

5x1 + 2x2 x3= 2y1 + 0y2 2y3 2

b )Now we realize that the vectors with xi terms and yi terms can be expressed as a matrix-vector product. Then we can express the system of equations in matrix form as 2 1 5 (b) Ans: 2 x1 5 3 x2 = 6 1 x3 2 3 4 2 3 y1 4 y2 + c 2 y3 2 4 0

2x1 + 3x2 2x3 5y1 2y2 + 3y3 = 200 x1 + 4x2 + 3x3 6y1 + 4y2 4y3 = 120 5x1 + 2x2 x3 2y1 + 0y2 + 2y3 = 350 Now we can express the left hand side as a single matrix-vector product of a 3-by-6 matrix with a 6-vector 2 3 1 2 3 1 4 5 2 #16 (10 points) 5 6 2 2 4 0 3 4 2 x1 x2 x3 y1 y2 y3 200 = 120 350

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