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Difficulty: MediumMatrices and Matrix Operations

A community theater sells adult tickets for $15\$15 each and student tickets for $10\$10 each. The number of tickets sold for the evening performances on Friday and Saturday is represented by the matrix AA:

A=[804511060]A = \begin{bmatrix} 80 & 45 \\ 110 & 60 \end{bmatrix}

where the rows represent Friday and Saturday, respectively, and the columns represent adult and student tickets, respectively.

The ticket prices are represented by the matrix BB:

B=[1510]B = \begin{bmatrix} 15 \\ 10 \end{bmatrix}

The product matrix C=ABC = AB represents the total ticket revenue for each night. What is the element in the second row and first column of matrix CC, which represents the total ticket revenue, in dollars, for Saturday's performance?

Answer: 2250 dollars

Answer

The correct answer is 22502250 dollars, which represents the ticket revenue for Saturday's performance.
The correct answer is 22502250 dollars. The element in the second row and first column of the product matrix C=ABC = AB is obtained by multiplying the elements of the second row of matrix AA (Saturday's ticket sales: 110110 adult, 6060 student) by the corresponding elements of the first column of matrix BB (ticket prices: $15\$15 adult, $10\$10 student) and summing the products: 110×15+60×10=1650+600=2250110 \times 15 + 60 \times 10 = 1650 + 600 = 2250.

Step-by-Step Solution

1
Identify the row of matrix AA representing Saturday's sales and the column of matrix BB representing the prices.
The second row of matrix AA is [11060]\begin{bmatrix} 110 & 60 \end{bmatrix} (representing 110110 adult tickets and 6060 student tickets sold on Saturday). Matrix BB is [1510]\begin{bmatrix} 15 \\ 10 \end{bmatrix} (representing the prices of $15\$15 and $10\$10 respectively).
Since the question asks for Saturday's performance (the second row of matrix AA) and the revenue is calculated by multiplying sales by prices, we use these specific components for the matrix multiplication.
2
Calculate the dot product of Saturday's sales row and the price column.
110×15+60×10=1650+600=2250110 \times 15 + 60 \times 10 = 1650 + 600 = 2250.
This calculation determines the sum of the revenue from adult tickets (110×15=1650110 \times 15 = 1650) and student tickets (60×10=60060 \times 10 = 600) to yield the total revenue.

Key Concept

Matrix Multiplication
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Topics

Matrices and Matrix Operations

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