 # For the following Leslie matrix , find an approximate expression for the population distribution after n years , given that the initial population dis Cheyanne Leigh 2021-02-02 Answered
For the following Leslie matrix , find an approximate expression for the population distribution after n years , given that the initial population distribution is given by $X\left(0\right)=\left[\begin{array}{c}2000\\ 4000\end{array}\right],{L}^{n}=\left[\begin{array}{cc}0.8& 0.4\\ 1.2& 0\end{array}\right]$
Select the correct choice below and fill in the answer boxes to complete your choise.
a)$X\approx \left(\right)\left({\right)}^{n}\left[\begin{array}{c}1\\ \left(\right)\end{array}\right]$
b)$X\approx \left({\right)}^{n}\left[\begin{array}{c}1\\ \left(\right)\end{array}\right]$
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Step 1
Every matrix has 2 attributes, rows and columns, if a matrix A is represented as ${A}_{mxn}$, then m represents the number of rows and n represents the number of columns present in the matrix. In order to add two matrices, A and B, the number of columns of matrix A should be equal to the number of rows of matrix B.
Two or more matrices can be used to solve linear equations by equating, using row or column operations in order to reduce the matrix and there are a lot of other ways as well. Identity matrix is a special kind of matrix with only 1 on its diagonal elements.
Step 2
Given Leslie Matrix:
${L}^{n}=\left[\begin{array}{cc}0.8& 0.4\\ 1.2& 0\end{array}\right]$
The initial value population distribution is given by:
$X\left(0\right)=\left[\begin{array}{c}2000\\ 4000\end{array}\right]$
The population distribution during the nth time period is given by
${X}_{n}={L}^{n}X\left(0\right)$
${X}_{n}={\left[\begin{array}{cc}0.8& 0.4\\ 1.2& 0\end{array}\right]}^{n}\left[\begin{array}{c}2000\\ 4000\end{array}\right]$
$={\left[\begin{array}{cc}0.8& 0.4\\ 1.2& 0\end{array}\right]}^{n}2000\left[\begin{array}{c}1\\ \frac{1}{2}\end{array}\right]$
${X}_{n}=2000{\left[\begin{array}{cc}0.8& 0.4\\ 1.2& 0\end{array}\right]}^{n}\left[\begin{array}{c}1\\ \frac{1}{2}\end{array}\right]$
${X}_{n}=2000{\left[\begin{array}{cc}0.8& 0.4\\ 1.2& 0\end{array}\right]}^{n}\left[\begin{array}{c}1\\ 0.5\end{array}\right]$
Hence,the correct option is Option A.
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