I am doing revision on matrices and came across this question. The solution (the matrix provided below the question) is there. I am not sure how or why 180 is in the position (1,4) (row and column respectively).
Any help is much appreciated! Thank you
On
Perhaps to elaborate on this, you represent the process as a Markov chain, and the matrix models the transitions between the 4 possible states: eggs, larvae, pupae, adults. Each column represents the stage you are starting from, and each row - the stage you are ending up in. So each 1 larvae (2nd column) will end up with 0 eggs, 0 larvae, 0.2 pupae and 0 adults (hence the elemnts in column 2).
The first entry in the last column corresponds to how many eggs [first row] end up from 1 adult [last column] over 1 period - and the answer from your problem (as also pointed out by P) is 180.
The answer is in the first sentence:
Each female adult of a particular species of stick insects lays 180 eggs at the start of each month.
Therefore $E_{n+1}=180A_n\Longrightarrow \begin{pmatrix}E_{n+1}\\\vdots\end{pmatrix}=\begin{pmatrix}0&0&0&180\\\vdots&\vdots&\vdots&\vdots\end{pmatrix}\cdot\begin{pmatrix}\vdots\\A_n\end{pmatrix}$.