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Year 12 Specialist (Unit 3 & 4) Applications of matrices

Leslie matrices

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Questions
Question 1
162946

Consider the following Leslie matrix \( L \) and population matrix \( P_0 \)
\[
L = \begin{bmatrix}
0 & 0 & 18 \\
\frac{1}{3} & 0 & 0 \\
0 & \frac{1}{2} & 0
\end{bmatrix} \quad \text{and} \quad P_0 = \begin{bmatrix}
1800 \\
0 \\
0
\end{bmatrix}
\]
Find:
i) \( L P_0 \)
ii) \( L^2 P_0 \)
iii) \( L^3 P_0 \)
iv) Comment on the long form behaviour of \( L P_0, L^2 P_0 \) and \( L^3 P_0 \).

i) \(\left[\begin{array}{c}0 \\ 600 \\ 0\end{array}\right]\)
ii) \(\left[\begin{array}{c}0 \\ 0 \\ 300\end{array}\right]\)
iii) \(\left[\begin{array}{c}5400 \\ 0 \\ 0\end{array}\right]\)
iv) The population cycles through three states

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