Chapter 3.8, Problem 36PPS

(a)

To determine

Matrix to represent the transitions in city population and suburb population.

Answer to Problem 36PPS

Matrix for the transitions in city population and suburb population:

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

Explanation of Solution

Given information:

Diagram for annual percent migration between a city and its suburbs:

  

Calculation:

Let

c be the current number of people living in the city.

And

s be the current number of people living in the suburbs.

Now,

The number of people living in the city after 1 year,

  $0.95c+0.03s$

The number of people living in the suburbs after 1 year,

  $0.05c+0.97s$

Then

Take the coefficients put them in a matrix $P$ :

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

Obtain the above two equations for the population after 1 year by multiplying matrix $P$ with column matrix:

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

Therefore,

Matrix for the transitions in city population and suburb population:

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

(b)

To determine

Prediction for number of people will live in the suburbs next year.

Answer to Problem 36PPS

17839 people will live in suburbs next year.

Explanation of Solution

Given information:

Diagram for annual percent migration between a city and its suburbs:

  

The number of people currently living in the city,

  $c=16275$

The number of people currently living in the suburbs,

  $s=17552$

Calculation:

Plug-in the values of c and s into the equations from Part (a):

  $\left[\begin{array}{cc}0.95& 0.03\\ 0.05& 0.97\end{array}\right]\left[\begin{array}{c}c\\ s\end{array}\right]=\left[\begin{array}{cc}0.95& 0.03\\ 0.05& 0.97\end{array}\right]\left[\begin{array}{c}16275\\ 17552\end{array}\right]$

Thus,

The equations become

For people will live in city after 1 year:

  $0.95c+0.03s=0.95\left(16275\right)+0.03\left(17552\right)\approx 15987$

For people will live in suburbs after 1 year:

  $0.05c+0.97s=0.05\left(16275\right)+0.97\left(17552\right)\approx 17839$

Therefore,

After 1 year, 15987 people will live in the city and 17839 people will live in suburbs.

(c)

To determine

The number of people lived in the city last year using the inverse of the matrix from Part (b).

Answer to Problem 36PPS

16587 people lived in the city last year.

Explanation of Solution

Given information:

Diagram for annual percent migration between a city and its suburbs:

  

Calculation:

Calculate the inverse of a $2\times 2$ matrix:

  ${\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

Substitute the values:

  $\begin{array}{l}{\left[\begin{array}{cc}0.95& 0.03\\ 0.05& 0.97\end{array}\right]}^{-1}=\frac{1}{\left(0.95\right)\left(0.97\right)-\left(0.03\right)\left(0.05\right)}\left[\begin{array}{cc}0.97& -0.03\\ -0.05& 0.95\end{array}\right]\\ =1.08695652174\left[\begin{array}{cc}0.97& -0.03\\ -0.05& 0.95\end{array}\right]\\ =\left[\begin{array}{cc}1.0543478& -0.0326087\\ -0.0543478& 1.0326087\end{array}\right]\end{array}$

Now,

Obtain the identity matrix by multiplying original with inverse:

  $\begin{array}{l}I=A^{-1}A\\ =\left[\begin{array}{cc}0.95& 0.03\\ 0.05& 0.97\end{array}\right]\left[\begin{array}{cc}1.0543478& -0.0326087\\ -0.0543478& 1.0326087\end{array}\right]\\ =\left[\begin{array}{cc}0.95\left(1.0543478\right)+0.03\left(-0.0543478\right)& 0.95\left(-0.0326087\right)+0.03\left(1.0326087\right)\\ 0.05\left(1.0543478\right)+0.97\left(-0.0543478\right)& 0.05\left(-0.0326087\right)+0.97\left(1.0326087\right)\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\end{array}$

Then

Use the inverse matrix in the same way the original matrix was used in Part (b).

Also, plug-in the values of c and s into the equation.

  $\left[\begin{array}{cc}1.0543478& -0.0326087\\ -0.0543478& 1.0326087\end{array}\right]\left[\begin{array}{c}c\\ s\end{array}\right]=\left[\begin{array}{c}1.0543478c+\left(-0.0326087\right)s\\ \left(-0.0543478\right)c+1.0326087s\end{array}\right]$

Thus,

The equation becomes

  $1.0543478c-0.0326087s=1.0543478\left(16275\right)-0.0326087\left(17552\right)\approx 16587$

Therefore,

16587 people lived in the city last year.