Chapter 12.1, Problem 26PPSE

Solution

A

a.

To determine: The total number of people participating in each activity by addition of the matrix.

The obtained matrix is $\left[\begin{array}{c}124.6\\ 113.3\\ 71.6\\ 87.2\end{array}\right]$

Given Information:

The table is defined,

Activity	Male	Female
Movies	59.2	65.4
Exercise Programs	54.3	59.0
Sports Events	40.5	31.1
Home Improvement	45.4	41.8

Explanation:

Consider the given table,

Activity	Male	Female
Movies	59.2	65.4
Exercise Programs	54.3	59.0
Sports Events	40.5	31.1
Home Improvement	45.4	41.8

Write given tables entries as matrix form:

  $\text{Male}=\left[\begin{array}{c}59.2\\ 54.3\\ 40.5\\ 45.4\end{array}\right]$ , and $\text{Female}=\left[\begin{array}{c}65.4\\ 59.0\\ 31.1\\ 41.8\end{array}\right]$

Add the both matrices.

  $\begin{array}{c}\text{Male}+\text{Female}=\left[\begin{array}{c}59.2\\ 54.3\\ 40.5\\ 45.4\end{array}\right]+\left[\begin{array}{c}65.4\\ 59.0\\ 31.1\\ 41.8\end{array}\right]\\ =\left[\begin{array}{c}59.2+65.4\\ 54.3+59.0\\ 40.5+31.1\\ 45.4+41.8\end{array}\right]\\ =\left[\begin{array}{c}124.6\\ 113.3\\ 71.6\\ 87.2\end{array}\right]\end{array}$

Thus, the total number of people participating in Movies is 124.6124.6 million, the total number of people participating in Exercise Programs is 113.3113.3 million, the total number of people participating in Sports Events is 71.671.6 million, and the total number of people participating in Home Improvement is 87.287.2 million.

Therefore, the obtained result is explained above.

b.

To determine: The difference between the numbers of people participating in each activity by subtraction of the matrix.

The obtained matrix is $\left[\begin{array}{c}-6.2\\ -4.7\\ 9.4\\ 3.6\end{array}\right]$

Given Information:

The table is defined,

Activity	Male	Female
Movies	59.2	65.4
Exercise Programs	54.3	59.0
Sports Events	40.5	31.1
Home Improvement	45.4	41.8

Explanation:

Consider the given table,

Activity	Male	Female
Movies	59.2	65.4
Exercise Programs	54.3	59.0
Sports Events	40.5	31.1
Home Improvement	45.4	41.8

Write given tables entries as matrix form:

  $\text{Male}=\left[\begin{array}{c}59.2\\ 54.3\\ 40.5\\ 45.4\end{array}\right]$ , and $\text{Female}=\left[\begin{array}{c}65.4\\ 59.0\\ 31.1\\ 41.8\end{array}\right]$

Subtract the both matrices.

  $\begin{array}{c}\text{Male}-\text{Female}=\left[\begin{array}{c}59.2\\ 54.3\\ 40.5\\ 45.4\end{array}\right]-\left[\begin{array}{c}65.4\\ 59.0\\ 31.1\\ 41.8\end{array}\right]\\ =\left[\begin{array}{c}59.2-65.4\\ 54.3-59.0\\ 40.5-31.1\\ 45.4-41.8\end{array}\right]\\ =\left[\begin{array}{c}-6.2\\ -4.7\\ 9.4\\ 3.6\end{array}\right]\end{array}$

Thus, the difference between the number of males and females participating in Movies is 6.26.2 million, the difference between the number of males and females participating in Exercise Programs is 4.74.7 million, the difference between the number of males and females participating in Sports Events is 9.49.4 million, and the difference between the number of males and females participating in Home Improvement is 3.63.6 million.

Therefore, the obtained result is explained above.

c.

To determine: The explanation for that the order of the matrices, matter in the part (b).

The order of the matrices do not matter.

Given Information:

The table is defined,

Activity	Male	Female
Movies	59.2	65.4
Exercise Programs	54.3	59.0
Sports Events	40.5	31.1
Home Improvement	45.4	41.8

Explanation:

Consider the given table,

The matrices are made by using the population data. And the population cannot be negative. In the part (b), the two first entries were negative. But the obtained values are absolute value the minus sign is discarded, as the number of people cannot be negative.

Only the absolute values of the differences are taken.

Therefore, the order of the matrices do not matter.