Cheyanne Leigh

2021-02-02

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)$X\approx \left({\right)}^{n}\left[\begin{array}{c}1\\ \left(\right)\end{array}\right]$

berggansS

Step 1
Every matrix has two characteristics: rows and columns. For example, if matrix A is shown as ${A}_{mxn}$, then the matrix's m rows and n columns are represented by the numbers m and n, respectively. The number of columns in matrix A must match the number of rows in matrix B in order to add the two matrices.
There are numerous methods for employing two or more matrices to solve linear equations, including equating them, reducing the matrix using row or column operations, and many others. A unique type of matrix called an identity matrix has only 1 on each of its diagonal members.
Step 2
Given Leslie Matrix:
${L}^{n}=\left[\begin{array}{cc}0.8& 0.4\\ 1.2& 0\end{array}\right]$
The distribution of initial values is provided by:
$X\left(0\right)=\left[\begin{array}{c}2000\\ 4000\end{array}\right]$
Given by is the population distribution for the nth time period.
${X}_{n}={L}^{n}X\left(0\right)$
${X}_{n}={\left[\begin{array}{cc}0.8& 0.4\\ 1.2& 0\end{array}\right]}^{n}\left[\begin{array}{c}2000\\ 4000\end{array}\right]$
$={\left[\begin{array}{cc}0.8& 0.4\\ 1.2& 0\end{array}\right]}^{n}2000\left[\begin{array}{c}1\\ \frac{1}{2}\end{array}\right]$
${X}_{n}=2000{\left[\begin{array}{cc}0.8& 0.4\\ 1.2& 0\end{array}\right]}^{n}\left[\begin{array}{c}1\\ \frac{1}{2}\end{array}\right]$
${X}_{n}=2000{\left[\begin{array}{cc}0.8& 0.4\\ 1.2& 0\end{array}\right]}^{n}\left[\begin{array}{c}1\\ 0.5\end{array}\right]$
Therefore, Option A is the best choice.

Jeffrey Jordon