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Difficulty: MediumMatrices and Matrix Operations
A clothing boutique sells two types of shirts: t-shirts and polo shirts. The number of shirts sold at the boutique's two locations, Downtown and Uptown, on a weekend is represented by matrix SS:
S = \begin{pmatrix} 40 & 30 \\ 50 & 20 \\end{pmatrix}
where the rows represent the locations (Downtown and Uptown, respectively) and the columns represent the shirt types (t-shirts and polo shirts, respectively).
The selling price and the production cost per shirt (in dollars) are represented by matrix CC:
C = \begin{pmatrix} 15 & 6 \\ 25 & 10 \\end{pmatrix}
where the rows represent the shirt types (t-shirts and polo shirts, respectively) and the columns represent the selling price and production cost, respectively.

Which of the following matrices represents the total selling revenue and total production cost for the two locations?

  1. \begin{pmatrix} 1350 & 540 \\ 1250 & 500 \\end{pmatrix}Answer
  2. B
    \begin{pmatrix} 600 & 180 \\ 1250 & 200 \\end{pmatrix}
  3. C
    \begin{pmatrix} 900 & 570 \\ 1500 & 950 \\end{pmatrix}
  4. D
    \begin{pmatrix} 1850 & 740 \\ 950 & 380 \\end{pmatrix}
  5. E
    \begin{pmatrix} 780 & 1300 \\ 870 & 1450 \\end{pmatrix}

Answer

The matrix representing the total selling revenue and total production cost for the two locations is the matrix with top row 1350 and 540, and bottom row 1250 and 500.
The correct matrix is obtained by multiplying the sales matrix SS by the price/cost matrix CC. Standard matrix multiplication pairs the quantity of each shirt type sold at each store with its respective selling price and production cost, yielding the correct total revenue and total production cost for both locations.

Step-by-Step Solution

1
Identify the dimensions and layout of the sales matrix SS and the price/cost matrix CC.
Matrix SS is a 2×22 \times 2 matrix representing sales. Matrix CC is a 2×22 \times 2 matrix representing price and cost. The product SCSC is defined because the number of columns in SS (2) equals the number of rows in CC (2).
Before multiplying, we must verify that the dimensions are compatible and that the product yields the desired real-world quantities.
2
Calculate the entries for the first row of the resulting matrix SCSC (Downtown location) by taking the dot product of the first row of SS with the columns of CC.
Revenue (Row 1, Column 1): 40×15+30×25=600+750=135040 \times 15 + 30 \times 25 = 600 + 750 = 1350.
Cost (Row 1, Column 2): 40×6+30×10=240+300=54040 \times 6 + 30 \times 10 = 240 + 300 = 540.
This yields the total revenue and production cost for the Downtown location by combining sales and prices/costs for both shirt types.
3
Calculate the entries for the second row of the resulting matrix SCSC (Uptown location) by taking the dot product of the second row of SS with the columns of CC.
Revenue (Row 2, Column 1): 50×15+20×25=750+500=125050 \times 15 + 20 \times 25 = 750 + 500 = 1250.
Cost (Row 2, Column 2): 50×6+20×10=300+200=50050 \times 6 + 20 \times 10 = 300 + 200 = 500.
This yields the total revenue and production cost for the Uptown location.

Key Concept

Matrix multiplication
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Topics

Matrices and Matrix Operations

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