A sporting goods company sells soccer balls and basketballs at two locations, North and So… — ACT

A sporting goods company sells soccer balls and basketballs at two locations, North and South. The matrix $A$ represents the inventory of these balls at each location:

A = \begin{bmatrix} 50 & 30 \\ 40 & 60 \end{bmatrix}

where the rows represent the locations (Row 1: North, Row 2: South) and the columns represent the ball types (Column 1: soccer balls, Column 2: basketballs).

The matrix $B$ represents the price, in dollars, of each type of ball under two different pricing structures (Column 1: regular price, Column 2: sale price):

B = \begin{bmatrix} 20 & 15 \\ 25 & 18 \end{bmatrix}

where Row 1 represents soccer balls and Row 2 represents basketballs.

Which of the following matrices represents the total revenue, in dollars, the company would collect from selling its entire inventory at each location under both pricing structures?

A
$\begin{bmatrix} 1,000 & 450 \\ 1,000 & 1,080 \end{bmatrix}$
B
$\begin{bmatrix} 1,600 & 1,500 \\ 1,970 & 1,830 \end{bmatrix}$
C
$\begin{bmatrix} 250 & 210 \\ 700 & 480 \end{bmatrix}$
$\begin{bmatrix} 1,750 & 1,290 \\ 2,300 & 1,680 \end{bmatrix}$ Answer
E
$\begin{bmatrix} 1,750 & 1,380 \\ 2,300 & 1,720 \end{bmatrix}$

Answer

\begin{bmatrix} 1,750 & 1,290 \\ 2,300 & 1,680 \end{bmatrix}

To find the total revenue under both pricing structures at each location, we perform the matrix multiplication

AB

. The entry in row

i

, column

j

of the resulting matrix corresponds to the revenue generated at location

i

under pricing structure

j

. Applying the row-by-column multiplication rule gives the correct matrix containing

1,750

1,290

2,300

, and

1,680

Step-by-Step Solution

Set up the matrix multiplication to compute the total revenue.

The total revenue matrix is represented by the product

AB

, where we multiply the

2 \times 2

inventory matrix

A

by the

2 \times 2

price matrix

B

Multiplying the quantities of items by their respective unit prices yields the total revenue.

Calculate the entries for the first row of the product matrix, representing the North location.

Row 1, Column 1 (North, Regular):

50(20) + 30(25) = 1,000 + 750 = 1,750

.
Row 1, Column 2 (North, Sale):

50(15) + 30(18) = 750 + 540 = 1,290

The first row of matrix

A

is multiplied by the columns of matrix

B

to find revenues under both pricing structures at the North location.

Calculate the entries for the second row of the product matrix, representing the South location.

Row 2, Column 1 (South, Regular):

40(20) + 60(25) = 800 + 1,500 = 2,300

.
Row 2, Column 2 (South, Sale):

40(15) + 60(18) = 600 + 1,080 = 1,680

The second row of matrix

A

is multiplied by the columns of matrix

B

to find revenues under both pricing structures at the South location.

Construct the final product matrix from the calculated values.

AB = \begin{bmatrix} 1,750 & 1,290 \\ 2,300 & 1,680 \end{bmatrix}

Placing the computed row values in their respective positions yields the completed revenue matrix.

Key Concept

Matrix multiplication is used to combine quantity data and unit cost data systematically across multiple categories and scenarios.

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