# Thread: transition matrix

1. ## transition matrix

A housekeeper buys three kinds of cereal: A, B, C. She never buys the same cereal in successive weeks. If she buys cereal A, then the next week she buys cereal B. However, if she buys either B or C, then the next week she is three times as likely to buy A as the other brand.
(a) Find the transition matrix.
(b) In the long run, how often does she buy each of the three brands?

not sure how toset this one up and get going

2. Math Forum - Ask Dr. Math

The above link might provide some foundational understanding.
The columns are brands A, B, and C respectively.
The rows are brands A, B, and C respectively.
We move from the column to the rows.
part a)

$
M_{3,3} =
\begin{bmatrix}
0 & \frac{3}{4} & \frac{3}{4} \\
1 & 0 & \frac{1}{4}\\
0 & \frac{1}{4} & 0 \\
\end{bmatrix}
$

part b)

Take powers of M and see if there are any trends.
$
M^n
$