A linear algebra homework assignment.Hope the teacher to guide the problem solving process and train of thought.

Writing Assignment 1 Math 222 April 24, 2020 Topic A sample of rock contains three radioactive isotopes: X, Y and Z. Every hour, the following happens: One-third of the atoms of X decay into Y . One-half of the atoms of Y decay into Z. Two-thirds of the atoms of Z decay into non-radioactive byproducts. Write down and diagonalize the \"transition matrix\" for this process of radioactive decay. In other words, find a matrix A M33 such that if x, y and z are the current amounts of the three isotopes, then x A y z gives the amounts of the isotopes one hour later. This is an example of a Markov process or Markov chain, an important application of linear algebra. 2 Format Your write-up should be no more than two pages. If you do not know how to typeset mathematical formulas, feel free to photograph and include pictures of your formulas in your write up. If you choose to handwrite the whole essay, ensure that your handwriting is very clear. The essay should be divided into three sections. 1. A brief introduction explaining what Markov processes are. 2. Your solution to the given problem. The transition matrix, with an explanation of how you obtained it and why it \"works\". Diagonalize the transition matrix. Demonstrate how to iterate applications of the transition matrix using the diagonalization. 3. Any concluding remarks you feel help explain the process, meaning or significance of your calculation. 4. A bibliography of resources your used to prepare the essay. 3 Grading The assignment is due on Saturday, May 9 2020, by 11:59pm (EDT). Due to deadlines for submitting grades, I will not accept submissions after that. Your essay will be graded on clarity, accuracy and overall quality. Proofread your work carefully to ensure a minimum of typos and grammatical errors. You can collaborate with your classmates and I will create a discussion forum to facilitate this. If you choose to turn in a group assignment, all participants must include their names on the assignment in order to get credit. The essay will be graded out of 10 points. Exceptionally good essays may be awarded up to 2 extra credit points. Since this essay is worth 5% of the course grade, that means that 1% extra credit for the entire course is available for this assignment. You may use any references you wish. Do not copy any portion of your essay from other sources. This is called plagiarism and is a form of academic dishonesty