Chapter 4.6, Problem 37PPS

a.

To determine

To write: a matrix to represent the transitions in player workshop.

Answer to Problem 37PPS

  $\begin{array}{l}\boxed{\text{From CD DAP}}\\ \boxed{\begin{array}{c}\text{To CD}\\ \text{To DAP}\end{array}}\left[\begin{array}{cc}0.35& 0.12\\ 0.65& 0.88\end{array}\right]\end{array}$

Explanation of Solution

Given information:

88% keep digital players.

12% switch to CD.

35% keep CD players.

65% switch to Digital.

Formula used:

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

Calculation:

According to the question,

Let x = the number of people who own CD players.

Let y = the number of people who own digital audio players.

Therefore, we get the following system of equations:

  $\begin{array}{l}0.35x+0.12y\\ 0.65x+0.88y\end{array}$

The matrix states that :

  $\begin{array}{l}\boxed{\text{From CD DAP}}\\ \boxed{\begin{array}{c}\text{To CD}\\ \text{To DAP}\end{array}}\left[\begin{array}{cc}0.35& 0.12\\ 0.65& 0.88\end{array}\right]\end{array}$

Hence, the matrix is formed.

b.

To determine

To calculate: The number of people who will own digital audio players next year.

Answer to Problem 37PPS

The total number of people that will own digital audio players next year are 20,218.

Explanation of Solution

Given information:

88% keep digital players.

12% switch to CD.

35% keep CD players.

65% switch to Digital.

Central City has a stable population of 25,000 people, of whom 17,252 own digital audio players and 7748 own portable CD players.

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 who will own CD players next year

Let y = the number of people who will own digital audio players next year.

  $\begin{array}{l}=\left[\begin{array}{cc}0.35& 0.12\\ 0.65& 0.88\end{array}\right]\left[\begin{array}{c}7748\\ 17,252\end{array}\right]=\left[\begin{array}{c}x\\ y\end{array}\right]\\ =\left[\begin{array}{c}\left(0.35\times 7748\right)+\left(0.12\times 17,252\right)\\ \left(0.65\times 7748\right)+\left(0.88\times 17,252\right)\end{array}\right]=\left[\begin{array}{c}x\\ y\end{array}\right]\\ =\left[\begin{array}{c}2711.8+2070.24\\ 5036.2+15,181.76\end{array}\right]=\left[\begin{array}{c}x\\ y\end{array}\right]\\ =\left[\begin{array}{c}4782.04\\ 20,217.96\end{array}\right]=\left[\begin{array}{c}x\\ y\end{array}\right]\end{array}$

Hence, the total number of people that will own digital audio players next year are 20,218.

c.

To determine

To calculate: an inverse matrix to find the number of people who owned digital audio players last year.

Answer to Problem 37PPS

4357 people owned digital audio players last year.

Explanation of Solution

Given information:

  $\left[\begin{array}{cc}0.35& 0.12\\ 0.65& 0.88\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}7748\\ 17,252\end{array}\right]$

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:

  $P=\left[\begin{array}{cc}0.35& 0.12\\ 0.65& 0.88\end{array}\right]$

Step 1.

  $=\left|\begin{array}{cc}0.35& 0.12\\ 0.65& 0.88\end{array}\right|=\left(0.35\times 0.88\right)-\left(0.12\times 0.65\right)$

  $\begin{array}{l}=0.308-0.078\\ =0.23\end{array}$

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

Step 2.

  $P^{-1}=\frac{1}{(0.35)(0.88)-(0.12)(0.65)}\left[\begin{array}{cc}0.88& -0.12\\ -0.65& 0.35\end{array}\right]$

  $\begin{array}{l}=\frac{1}{0.308-0.078}\left[\begin{array}{cc}0.88& -0.12\\ -0.65& 0.35\end{array}\right]\\ =\frac{1}{0.23}\left[\begin{array}{cc}0.88& -0.12\\ -0.65& 0.35\end{array}\right]\end{array}$

  $a=0.88,b=-0.12,c=-0.65,d=0.35$

Step3.

  $\begin{array}{l}=\frac{1}{0.23}\left[\begin{array}{cc}0.88& -0.12\\ -0.65& 0.35\end{array}\right]\left[\begin{array}{cc}0.35& 0.12\\ 0.65& 0.35\end{array}\right].\left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{0.23}\left[\begin{array}{cc}0.88& -0.12\\ -0.65& 0.35\end{array}\right]\left[\begin{array}{c}7748\\ 17,252\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{0.23}\left[\begin{array}{c}(0.88\times 7748)+(-0.12\times 17,252)\\ (-0.65\times 7748)+(0.35\times 17,252)\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{0.23}\left[\begin{array}{c}6818.24-2070.24\\ -5036.2+6038.2\end{array}\right]\\ =\left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{0.23}\left[\begin{array}{c}4748\\ 1002\end{array}\right]\end{array}$

Or

  $=\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}20,643\\ 4357\end{array}\right]$

Hence, about 4357 people owned digital audio players last year.