Dry Cleaning The dry cleaning example in the text used the following transition matrix:

Johnson NorthClean

Johnson $\left[\begin{array}{cc}0\mathrm{.}8& 0\mathrm{.}2\\ 0\mathrm{.}35& 0\mathrm{.}65\end{array}\right]$

Suppose now that each customer brings in one batch of clothes per week. Use various powers of the transition matrix to find the probability that a customer initially bringing a batch of clothes to Johnson also brings a batch to Johnson after the following time periods.

a$.1$ week

b$.2$ weeks

c$.3$ weeks

d$.4$ weeks

e$.$ What is the probability that a customer initially bringing a batch of clothes to NorthClean brings a batch to Johnson after $2$ weeks?