A refinery blends four petroleum components into three grades of gasoline

A refinery blends four petroleum components into three grades of gasoline - regular, premium and low lead.... (See attached file for full problem description)

Finding the eigenvalues and eigenvectors of a linear transformation represented by a 3 x 3 matrix, and using them to diagonalise the matrix.

Please show all steps and workings clearly and explain in detail how you derive the answers. 1) This question concerns the linear transformation represented by a 3x3 matrix of real numbers. With respect to the standard basis in both the domain and codomain. (a) Determine the characteristic polynomial for A. [2] (b) Show that lambda = 1 is an eigenvalue and find the remaining eigenvalues of A. [3] (c) Determine all the corresponding eigenvectors and eigenspaces. [6] (d) Write down a matrix P and a diagonal matrix D such that (P^-1)AP=D [3] (e) Hence verify that (P^-1)AP=D by matrix multiplication. [6]

What would be the difficulties associated with modeling this situation?

After contacting all of the professionals, you set up a "blog" (weblog) for you and them to share. You'll use this "blog" as the weeks go by to ask the group questions, raise issues for discussion and encourage networking among this diverse group. You decide to start your article with a math example that the readers can understand. Describe a real world situation that could be modeled by a function that is increasing, then constant, then decreasing. What would be the difficulties associated with modeling this situation?

Understanding integer programming models

A boat company manufactures three kinds of boats. Bass boats, ski boats and speed boats. Profit: Bass boat = 20,500 Ski boat = 12,000 Speed boat = 22,300 The company believes it will sell more bass boats than the other two boats combined, but no more than twice as many. Ski boats are it's standard model, bass boats and speed boats are modifications. The company has a production capacity to manufacture 210 standard boats. A bass boat requires 1.3 times the standard production capacity. A speed boat requires 1.5 times the normal production capacity. Only 160 of the high powered engines that are installed in the bass boats and 2 of those installed in the speed boats are available. ... click for more

Markov process and random walk

a markov process described by: T=[1/2 1/3 0 0;1/3 1/3 1/3 0;0 1/3 1/3 1/3;0 0 1/2 1/2] 1) what is the spatial interpretation of that markov process in terms of a random walk on {1,2,3,4}? 2) if a vector v=[1,0,0,0], and I want to compute v(T^m) for m=1,5,10 and 40. How do I express the resulting vectors in terms of an eigenvector of T or T transpose? and if I change the vector v, will the eigenvalues and eigenvector change?