Problem source

It has been observed that Professor Ecks proves three types of theorems:
1, those that are correct and new; 2, those that are correct, but
already known; 3, those that are false. It has also been observed
that, if a certain of her theorems is of type $i$, then her next
theorem is of type $j$ with probability $p\low_{ij},$ where $p\low_{ij}$
is the entry in the $i$th row and $j$th column of the following
array: 
\[
\begin{pmatrix}0.3 & 0.3 & 0.4\\
0.2 & 0.4 & 0.4\\
0.1 & 0.3 & 0.6
\end{pmatrix}\,.
\]
Let $a_{i},$ $i=1,2,3$, be the probability that a given theorem
is of type $i$, and let $b_{j}$ be the consequent probability that
the next theorem is of type $j$. 
\begin{questionparts}
\item Explain why $b_{j}=a\low_{1}p\low_{1j}+a\low_{2}p\low_{2j}+a\low_{3}p\low_{3j}\,.$
\item Find values of $a\low_{1},a\low_{2}$ and $a\low_{3}$ such that $b_{i}=a_{i}$
for $i=1,2,3.$
\item For these values of the $a_{i}$ find the probabilities $q\low_{ij}$
that, if a particular theorem is of type $j$, then the \textit{preceding
}theorem was of type $i$. 
\end{questionparts}