A sporting goods chain has two locations. Location 1 sells 10 soccer balls and 15 basketba… — ACT

A sporting goods chain has two locations. Location 1 sells 10 soccer balls and 15 basketballs. Location 2 sells 8 soccer balls and 20 basketballs. This is represented by the sales matrix $A = \begin{bmatrix} 10 & 15 \\ 8 & 20 \\end{bmatrix}$ . Soccer balls cost $15 each and basketballs cost$ 25 each, represented by the price matrix $B = \begin{bmatrix} 15 \\ 25 \\end{bmatrix}$ . What is the product $AB$ , which represents the total revenue generated by each location?

A
$\begin{bmatrix} 150 \\ 500 \\end{bmatrix}$
B
$\begin{bmatrix} 350 \\ 725 \\end{bmatrix}$
C
$\begin{bmatrix} 525 \\ 520 \\end{bmatrix}$
$\begin{bmatrix} 525 \\ 620 \\end{bmatrix}$ Answer
E
$\begin{bmatrix} 525 & 620 \\end{bmatrix}$

Answer

The column matrix with elements 525 in the first row and 620 in the second row

The correct answer is the column matrix containing 525 and 620. This is found by multiplying each row of the sales matrix (representing each location) by the column of the price matrix (representing prices of the items):

10(15) + 15(25) = 525

for the first row, and

8(15) + 20(25) = 620

for the second row.

Step-by-Step Solution

Set up the matrix multiplication of the

2 \times 2

sales matrix

A

and the

2 \times 1

price matrix

B

AB = \begin{bmatrix} 10 & 15 \\ 8 & 20 \\end{bmatrix} \begin{bmatrix} 15 \\ 25 \\end{bmatrix}

To find the total revenue for each location, we multiply the sales of each item by its corresponding price.

Multiply the first row of

A

by the column of

B

to find the element in the first row of the product matrix (revenue for Location 1).

10(15) + 15(25) = 150 + 375 = 525

This calculates the total revenue for Location 1 by summing the revenue from soccer balls (

10 \times 15

) and basketballs (

15 \times 25

Multiply the second row of

A

by the column of

B

to find the element in the second row of the product matrix (revenue for Location 2).

8(15) + 20(25) = 120 + 500 = 620

This calculates the total revenue for Location 2 by summing the revenue from soccer balls (

8 \times 15

) and basketballs (

20 \times 25

Combine the results into a

2 \times 1

resulting matrix.

\begin{bmatrix} 525 \\ 620 \\end{bmatrix}

Since matrix

A

has dimensions

2 \times 2

and matrix

B

has dimensions

2 \times 1

, their product

AB

must have dimensions

2 \times 1

Key Concept

Matrix multiplication involves multiplying the rows of the first matrix by the columns of the second matrix, matching corresponding elements and adding the products.

Alternative Method

Instead of matrix multiplication, the revenue can be computed using standard arithmetic: Location 1 revenue is

10 \times \$15 + 15 \times \$25 = \$525

and Location 2 revenue is

8 \times \$15 + 20 \times \$25 = \$620

. The resulting column matrix is

\begin{bmatrix} 525 \\ 620 \\end{bmatrix}

Estimated Time:1m 0s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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