Chapter 2.3, Problem 50E

a.

To determine

To find: the difference in arts patronage from 1992 to 2002 in the form of matrix.

Answer to Problem 50E

  $\left[\begin{array}{ccc}\begin{array}{l}8\\ 6\\ 6\\ 5\end{array}& \begin{array}{l}7\\ 4\\ 3\\ 2\end{array}& \begin{array}{l}5\\ 4\\ 1\\ 2\end{array}\end{array}\right]$

Explanation of Solution

Given information: The matrix showing the percentage of people listening or watching performances in 1992 and 2002 is as follows:

  $\begin{array}{cc}1992& \begin{array}{ccc}\text{TV}& \text{Radio}& \text{Recording}\end{array}\\ \begin{array}{c}\text{Classical}\\ \text{Jazz}\\ \text{Opera}\\ \text{Musical}\end{array}& \left[\begin{array}{ccc}\begin{array}{l}26\\ 22\\ 12\\ 17\end{array}& \begin{array}{l}31\\ 28\\ 9\\ 4\end{array}& \begin{array}{l}24\\ 21\\ 7\\ 6\end{array}\end{array}\right]\end{array}\begin{array}{cc}2002& \begin{array}{ccc}\text{TV}& \text{Radio}& \text{Recording}\end{array}\\ \begin{array}{c}\text{Classical}\\ \text{Jazz}\\ \text{Opera}\\ \text{Musical}\end{array}& \left[\begin{array}{ccc}\begin{array}{l}18\\ 16\\ 6\\ 12\end{array}& \begin{array}{l}24\\ 24\\ 6\\ 2\end{array}& \begin{array}{l}19\\ 17\\ 6\\ 4\end{array}\end{array}\right]\end{array}$

Calculation:

Subtract the matrix for 2002 from 1992 and solve to get difference in arts patronage during this time as shown below.

  $\begin{array}{c}\left[\begin{array}{ccc}\begin{array}{l}26\\ 22\\ 12\\ 17\end{array}& \begin{array}{l}31\\ 28\\ 9\\ 4\end{array}& \begin{array}{l}24\\ 21\\ 7\\ 6\end{array}\end{array}\right]-\left[\begin{array}{ccc}\begin{array}{l}18\\ 16\\ 6\\ 12\end{array}& \begin{array}{l}24\\ 24\\ 6\\ 2\end{array}& \begin{array}{l}19\\ 17\\ 6\\ 4\end{array}\end{array}\right]=\left[\begin{array}{ccc}\begin{array}{l}26-18\\ 22-16\\ 12-6\\ 17-12\end{array}& \begin{array}{l}31-24\\ 28-24\\ 9-6\\ 4-2\end{array}& \begin{array}{l}24-19\\ 21-17\\ 7-6\\ 6-4\end{array}\end{array}\right]\\ =\left[\begin{array}{ccc}\begin{array}{l}8\\ 6\\ 6\\ 5\end{array}& \begin{array}{l}7\\ 4\\ 3\\ 2\end{array}& \begin{array}{l}5\\ 4\\ 1\\ 2\end{array}\end{array}\right]\end{array}$

The resultant matrix shows the decrease in each area during this duration.

b.

To determine

the area that has greatest decrease from 1992 to 2002.

Answer to Problem 50E

Classical music on TV

Explanation of Solution

Given information:

The matrix showing the decrease in percentage of people listening or watching performances during 1992 to 2002 is as follows:

  $\begin{array}{cc}1992-2002& \begin{array}{ccc}\text{TV}& \text{Radio}& \text{Recording}\end{array}\\ \begin{array}{c}\text{Classical}\\ \text{Jazz}\\ \text{Opera}\\ \text{Musical}\end{array}& \left[\begin{array}{ccc}\begin{array}{l}8\\ 6\\ 6\\ 5\end{array}& \begin{array}{l}7\\ 4\\ 3\\ 2\end{array}& \begin{array}{l}5\\ 4\\ 1\\ 2\end{array}\end{array}\right]\end{array}$

Calculation:

Look at the resultant matrix. The first row and first column of this matrix has greatest numbers. So, the greatest number shows the area that saw the greatest decrease in this time.

Since the first row and first column represents classical music and TV respectively. So, the classical music on TV saw a greatest decrease.