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Difficulty: MediumMatrices and Matrix Operations

A cleaning service company offers two types of packages: Standard Clean and Deep Clean. The number of packages sold during the first two weeks of the month is represented by matrix QQ:

Q=[20152518]Q = \begin{bmatrix} 20 & 15 \\ 25 & 18 \end{bmatrix}

where row 1 represents Week 1, row 2 represents Week 2, column 1 represents Standard Clean packages, and column 2 represents Deep Clean packages.

The price charged and the cost to perform each package (in dollars) are represented by matrix PP:

P=[803015060]P = \begin{bmatrix} 80 & 30 \\ 150 & 60 \end{bmatrix}

where row 1 represents Standard Clean packages, row 2 represents Deep Clean packages, column 1 represents the price charged, and column 2 represents the cost.

Which of the following matrices represents the total revenue and total cost for Week 1 and Week 2, respectively?

  1. A
    [1,6004503,7501,080]\begin{bmatrix} 1,600 & 450 \\ 3,750 & 1,080 \end{bmatrix}
  2. B
    [2,3501,7404,5003,330]\begin{bmatrix} 2,350 & 1,740 \\ 4,500 & 3,330 \end{bmatrix}
  3. [3,8501,5004,7001,830]\begin{bmatrix} 3,850 & 1,500 \\ 4,700 & 1,830 \end{bmatrix}Answer
  4. D
    [3,8001,5004,0001,830]\begin{bmatrix} 3,800 & 1,500 \\ 4,000 & 1,830 \end{bmatrix}
  5. E
    [650300700330]\begin{bmatrix} 650 & 300 \\ 700 & 330 \end{bmatrix}

Answer

The matrix with elements 3,850 and 1,500 in the first row, and 4,700 and 1,830 in the second row.
The matrix representing the total revenue and total cost for Week 1 and Week 2 is found by multiplying matrix QQ by matrix PP. Since QQ is a 2×22 \times 2 matrix (weeks by package types) and PP is a 2×22 \times 2 matrix (package types by financial categories), the product Q×PQ \times P yields a 2×22 \times 2 matrix representing weeks by financial categories (revenue and cost).

Calculating each element:
- Row 1, Column 1 (Week 1 Revenue): (20×80)+(15×150)=1,600+2,250=3,850(20 \times 80) + (15 \times 150) = 1,600 + 2,250 = 3,850
- Row 1, Column 2 (Week 1 Cost): (20×30)+(15×60)=600+900=1,500(20 \times 30) + (15 \times 60) = 600 + 900 = 1,500
- Row 2, Column 1 (Week 2 Revenue): (25×80)+(18×150)=2,000+2,700=4,700(25 \times 80) + (18 \times 150) = 2,000 + 2,700 = 4,700
- Row 2, Column 2 (Week 2 Cost): (25×30)+(18×60)=750+1,080=1,830(25 \times 30) + (18 \times 60) = 750 + 1,080 = 1,830

This results in the matrix:
[3,8501,5004,7001,830]\begin{bmatrix} 3,850 & 1,500 \\ 4,700 & 1,830 \end{bmatrix}

Step-by-Step Solution

1
Identify the appropriate matrix operation to combine the weekly package sales with their respective price and cost.
We need to compute the product of matrix QQ and matrix PP (Q×PQ \times P), since multiplying a 2×22 \times 2 matrix of (weeks ×\times packages) by a 2×22 \times 2 matrix of (packages ×\times financial category) results in a 2×22 \times 2 matrix of (weeks ×\times financial category).
This alignment ensures that the package types cancel out, leaving the desired weekly revenue and cost breakdown.
2
Calculate the elements of the first row of the product matrix.
Row 1, Column 1 (Week 1 Revenue): 20(80)+15(150)=1,600+2,250=3,85020(80) + 15(150) = 1,600 + 2,250 = 3,850. Row 1, Column 2 (Week 1 Cost): 20(30)+15(60)=600+900=1,50020(30) + 15(60) = 600 + 900 = 1,500.
The first row represents Week 1 revenue and cost.
3
Calculate the elements of the second row of the product matrix.
Row 2, Column 1 (Week 2 Revenue): 25(80)+18(150)=2,000+2,700=4,70025(80) + 18(150) = 2,000 + 2,700 = 4,700. Row 2, Column 2 (Week 2 Cost): 25(30)+18(60)=750+1,080=1,83025(30) + 18(60) = 750 + 1,080 = 1,830.
The second row represents Week 2 revenue and cost.

Key Concept

Matrix multiplication represents linear combinations of variables, useful for calculating total cost and revenue in multi-variable business scenarios.
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Topics

Matrices and Matrix Operations

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