How do you multiply transition probability matrixes? Say I'm trying to find $P^{2}$ and I just need to multiply the initial state (this is the middle row) \begin{bmatrix}0.25&0.5&0.25\end{bmatrix} by the transition probability matrix
\begin{bmatrix}0.6&0.25&0.15\\0.25&0.5&0.25\\0.18&.52&.3\end{bmatrix}
The answer should be \begin{bmatrix}0.32&0.4425&0.2375\end{bmatrix}
I just can't figure out what exactly I'm supposed to be multiplying that would give me that answer.
edit: The solution I was given lists the answer as that first intitial state/line vector times the whole matrix, and the answer is as I listed.  Here is the full question, but its just how to go from one step to another that I can't figure out: 
Suppose that the offspring’s
choice of post-secondary education
(university/college/trade) is dependent on the highest level of education
received by their parents. If one parent went to university, there is a 60%
chance that the offspring
will attend university, 25% chance that they will
attend college. If one parent went to college, there is a 50% chance they
will attend college as well, and 25% chance they will enter a trade. If one
parent works in a trade, there is a 52% chance that they will attend
college, and 30% chance they will enter a trade.
If a child currently has one parent who chose to go to college, what are
the chances that the child’s future child will also attend college?
 A: You are getting muddled in your notation.
Usually the $n$th state is given by $x^{(n)}$ and the transition matrix is given by $P$.
You will have started with $x^{(0)}=\begin{bmatrix}0&1&0\ \end{bmatrix}$
You then worked out $x^{(1)}=x^{(0)}P=\begin{bmatrix}0&1&0\ \end{bmatrix} \begin{bmatrix}0.6&0.25&0.15\\0.25&0.5&0.25\\0.18&.52&.3\end{bmatrix}
 = \begin{bmatrix}0.25&0.5&0.25\ \end{bmatrix}$
If you don't really understand matrix multiplication then you won't understand how you got there, so I will try to explain.
To get the first element of $x^{(1)}$ you multiply the elements of the first column of the matrix by the corresponding elements of $x^{(0)}$ and add them together:
$0.6 \times 0 + 0.25 \times 1  + 0.18 \times 0 = 0.25$
$0.25 \times 0 + 0.5 \times 1  + 0.52 \times 0 = 0.52$
$0.15 \times 0 + 0.25 \times 1  + 0.3 \times 0 = 0.25$
What you want next is not $P^2$ - that would be the $3 \times 3$ matrix representing the transition over two generations.
Instead you want $x^{(2)}=x^{(1)}P$
$x^{(2)}=\begin{bmatrix}0.25&0.5&0.25\ \end{bmatrix} \begin{bmatrix}0.6&0.25&0.15\\0.25&0.5&0.25\\0.18&.52&.3\end{bmatrix}$
Now follow the same procedure as before. I'll do the first element for you:
$0.6 \times 0.25 + 0.25 \times 0.5  + 0.18 \times 0.25 = 0.32$
