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?
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:
$0.6 \times 0 + 0.25 \times 1 + 0.18 \times 0 = 0.25$
$0.25 \times 0 + 0.5 \times 1 + 0.52 \times 0 = 0.5$
$0.15 \times 0 + 0.25 \times 1 + 0.3 \times 0 = 0.25$
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:
$0.6 \times 0.25 + 0.25 \times 0.5 + 0.18 \times 0.25 = 0.32$