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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

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up vote 2 down vote accepted

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 $E_{n+1}=180A_n\Longrightarrow \begin{pmatrix}E_{n+1}\\\vdots\end{pmatrix}=\begin{pmatrix}0&0&0&180\\\vdots&\vdots&\vdots&\vdots\end{pmatrix}\cdot\begin{pmatrix}\vdots\\A_n\end{pmatrix}$.

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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.

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