Chapter 4.6, Problem 36PPS

a.

To determine

To write: a matrix to represent the transitions in city population and suburb population.

Answer to Problem 36PPS

  $\begin{array}{l}\boxed{\text{From city suburbs}}\\ \boxed{\begin{array}{c}\text{To city}\\ \text{To suburbs}\end{array}}\left[\begin{array}{cc}0.95& 0.03\\ 0.05& 0.97\end{array}\right]\end{array}$

Explanation of Solution

Given information:

The annual percentage migration from city to city is 0.95.

The annual percentage migration from city to suburbs is 0.05.

The annual percentage migration from suburbs to city is 0.03.

The annual percentage migration from suburbs to suburbs is 0.97.

Formula used:

A matrix is a rectangular array of variables or constants in horizontal rows and vertical columns, enclosed in brackets.

Calculation:

Let x = the number of people who live in city.

Let y = the number of people who live in suburbs.

Therefore, we get the following system of equations:

  $\begin{array}{l}0.95x+0.03y\\ 0.05x+0.97y\end{array}$

The matrix states that :

  $\begin{array}{l}\boxed{\text{From city suburbs}}\\ \boxed{\begin{array}{c}\text{To city}\\ \text{To suburbs}\end{array}}\left[\begin{array}{cc}0.95& 0.03\\ 0.05& 0.97\end{array}\right]\end{array}$

Hence, the matrix is formed.

b.

To determine

To calculate: The number of people who will live in the suburbs next year.

Answer to Problem 36PPS

The total number of people living in suburbs next year will be 17,839.

Explanation of Solution

Given information:

The annual percentage migration from city to city is 0.95.

The annual percentage migration from city to suburbs is 0.05.

The annual percentage migration from suburbs to city is 0.03.

The annual percentage migration from suburbs to suburbs is 0.97.

Currently, there are 16,275 people living in the city and 17, 552 people living in the suburbs.

Formula used:

Multiplying Matrices : $$

  $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right].\left[\begin{array}{cc}e& f\\ g& h\end{array}\right]=\left[\begin{array}{cc}ae+bg& af+bh\\ ce+dg& cf+dh\end{array}\right]$

Calculation:

According to the question we have,

Let x = the number of people living in the city

Let y = the number of people living in the suburbs.

  $\begin{array}{l}=\left[\begin{array}{cc}0.95& 0.03\\ 0.05& 0.97\end{array}\right]\left[\begin{array}{c}16,275\\ 17,552\end{array}\right]=\left[\begin{array}{c}x\\ y\end{array}\right]\\ =\left[\begin{array}{c}\left(0.95\times 16,275\right)+\left(0.03\times 17,552\right)\\ \left(0.05\times 16,275\right)+\left(0.97\times 17,552\right)\end{array}\right]=\left[\begin{array}{c}x\\ y\end{array}\right]\\ =\left[\begin{array}{c}15461.25+526.56\\ 813.75+17,025.44\end{array}\right]=\left[\begin{array}{c}x\\ y\end{array}\right]\\ =\left[\begin{array}{c}15987.81\\ 17839.19\end{array}\right]=\left[\begin{array}{c}x\\ y\end{array}\right]\end{array}$

Hence, the total number of people living in suburbs next year will be 17,839.

c.

To determine

To calculate: a inverse matrix to find the number of people lived in the city last year.

Answer to Problem 36PPS

16587people lived in the city last year.

Explanation of Solution

Given information:

  $\left[\begin{array}{cc}0.95& 0.03\\ 0.05& 0.97\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}16,275\\ 17,552\end{array}\right]$

Formula used:

Inverse of a $2\times 2$ matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is $A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$ is, where $ad-bc\ne 0$ .

The value of a second order determinant is the difference of the products of the products of the two diagonals.

Steps to solve the equations are:-

Step1. Find the inverse of the coefficient matrix.

Step 2. Multiple each side of the matrix equation by the inverse matrix.

Calculation:

  $P=\left[\begin{array}{cc}0.95& 0.03\\ 0.05& 0.97\end{array}\right]$

Step 1.

  $=\left|\begin{array}{cc}0.95& 0.03\\ 0.05& 0.97\end{array}\right|=\left(0.95\times 0.97\right)-\left(0.03\times 0.05\right)$

  $\begin{array}{l}=0.9215-0.0015\\ =0.92\end{array}$

Since the determinant does not equal to 0, $P^{-1}$ exists.

Step 2.

  $P^{-1}=\frac{1}{(0.95)(0.97)-(0.03)(0.05)}\left[\begin{array}{cc}0.97& -0.03\\ -0.05& 0.95\end{array}\right]$

  $\begin{array}{l}=\frac{1}{0.9215-0.0015}\left[\begin{array}{cc}0.97& -0.03\\ -0.05& 0.95\end{array}\right]\\ =\frac{1}{0.92}\left[\begin{array}{cc}0.97& -0.03\\ -0.05& 0.95\end{array}\right]\end{array}$

  $a=0.97,b=-0.03,c=-0.05,d=0.95$

Step3.

  $\begin{array}{l}=\frac{1}{0.92}\left[\begin{array}{cc}0.97& -0.03\\ -0.05& 0.95\end{array}\right]\left[\begin{array}{cc}0.95& 0.03\\ 0.05& 0.97\end{array}\right].\left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{0.92}\left[\begin{array}{cc}0.97& -0.03\\ -0.05& 0.95\end{array}\right]\left[\begin{array}{c}16,275\\ 17,552\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{0.92}\left[\begin{array}{c}(0.97\times 16,275)+(-0.03\times 17,552)\\ (-0.05\times 16,275)+(0.95\times 17,252)\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{0.92}\left[\begin{array}{c}15786.75-526.56\\ -813.75+16389.4\end{array}\right]\\ =\left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{0.92}\left[\begin{array}{c}15260.19\\ 15,575.65\end{array}\right]\end{array}$

Or

  $=\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}16,587.16304\\ 16,930.05435\end{array}\right]$

Hence, about 16587 people lived in the city last year.