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Study Lesson 3 (Matrix Math), Lesson 4 (The Inverse of a Matrix and Applications), Lesson 5 (Elementary Matrices), and Lesson 14 (Markov Analysis) from my Linear Algebra & Vector Geometry book to prepare for this assignment.

This is just classic matrix math stuff as taught in Lesson 3 of my book.

Classic inverse matrix stuff as taught in Lesson 4 of my book.  Note,  that the row-reduction method will "determine whether A is invertible."  Simply put, if you are able to row-reduce A down to an Identity matrix, A is invertible.  If not, it is not.

You can also use determinants to determine whether a matrix is invertible or not as I teach in Lesson 6, but it is apparent that you are not supposed to know that, yet.  That lesson will be included in Assignment 5.

Note that you can verify that your answer is correct by confirming that the product of A and A-inverse is the Identity matrix.

Follow instructions.  You can find the inverse of the coefficient matrix and then solve the system, similar to my Lesson 4, question 1.  It is unclear whether you are allowed to use the shortcut to find the inverse of a 2 by 2 matrix I teach in Lesson 3. 

Personally, I wouldn't take chances.  Use the row-reduction method to find the inverse, just in case they penalize you for using a determinant shortcut that may not be taught until later.  Certainly, by the time you write your final exam, you would be allowed to use any method to find an inverse unless specifically instructed to use one technique in a question.

This is a Markov Analysis question similar to my questions in Lesson 14. 

In part (a), trust their notation.  By X(2), they mean after 2 time periods even though they say the "third observation period."  The first is X(0), the initial state; the second is X(1), and so, the third is X(2).

When you are finding the steady-state or stable vector, don't worry about the little tricks I use.  Set up the augmented matrix with a row of ones all the way through the first row, then the rest of the augmented matrix is I - T for the coefficients augmented with a column of zeros.  In other words, do Step 1 as I outline at the start of Lesson 14.  At that point, merely row-reduce the way you always do.  Don't worry about the fancy tricks I show about making zero rows and stuff.  Row-reduce like usual, and the system will solve itself.  I really regret over-complicating this lesson by giving too many tricks when it is ultimately just a row-reduction problem. 

DO NOT USE THE CRAMER'S RULE METHOD I DEMONSTRATE!  Obviously, since that isn't even taught until later in the course (Lesson 8 in my book).  Even after you do learn Lesson 8, do not use this approach to solve Markov.  Unnecessary.  Stick to good old row-reduction.