Question

the fee structure of a dance class is given below:

\

$$\begin{tabular}{|c|c|c|} \\hline timing & adult & children \\\\ \\hline before 10 a.m. & \\$45 & \\$60 \\\\ \\hline 10 a.m. to 4 p.m. & \\$100 & \\$150 \\\\ \\hline after 4 p.m. & \\$125 & \\$175 \\\\ \\hline \\end{tabular}$$

write a matrix to represent the given data.

\\( p = \

$$\begin{bmatrix} 45 & 60 \\\\ 100 & 150 \\\\ 125 & 175 \\end{bmatrix}$$

, q = \

$$\begin{bmatrix} 45 & 100 & 125 \\\\ 60 & 150 & 175 \\end{bmatrix}$$

\\)

\\( p = \

$$\begin{bmatrix} 45 & 60 \\\\ 150 & 100 \\\\ 125 & 175 \\end{bmatrix}$$

, q = \

$$\begin{bmatrix} 45 & 150 & 125 \\\\ 60 & 100 & 175 \\end{bmatrix}$$

\\)

Explanation:

Identify the table structure and elements

The given table has 3 rows representing timing categories and 2 columns representing age groups:

• Row 1 (Before 10 A.M.): Adult = 45, Children = 60
• Row 2 (10 A.M. to 4 P.M.): Adult = 100, Children = 150
• Row 3 (After 4 P.M.): Adult = 125, Children = 175

Construct the standard matrix representation

Representing rows as timings and columns as age groups yields a \(3 \times 2\) matrix \(P\):

$$ P =
LATEXBLOCK0
$$

Construct the transposed matrix representation

Representing rows as age groups and columns as timings yields a \(2 \times 3\) matrix \(Q\):

$$ Q =
LATEXBLOCK1
$$

Comparing this with the choices, the first option correctly lists these two representations.

Answer:

• **(A) \(P =

$$\begin{bmatrix} 45 & 60 \\ 100 & 150 \\ 125 & 175 \end{bmatrix}$$

, Q =

$$\begin{bmatrix} 45 & 100 & 125 \\ 60 & 150 & 175 \end{bmatrix}$$

\) (Correct answer)**

• (B) \(P =

$$\begin{bmatrix} 45 & 60 \\ 150 & 100 \\ 125 & 175 \end{bmatrix}$$

, Q =

$$\begin{bmatrix} 45 & 150 & 125 \\ 60 & 100 & 175 \end{bmatrix}$$

\)