Question

a store sells drinks in 3 sizes. the matrix shows how many of each size cup are ordered for the first 3 months of the year.
jan feb mar
sm 150 175 100
med 150 150 125
lg 125 250 300
the cost of a small cup is $0.03, a medium cup is $0.035, and a large cup is $0.04.
which matrix multiplication could be used to find out how much is spent on cups for the first three months?
a. \\( \

$$\begin{bmatrix} 0.03 \\\\ 0.035 \\\\ 0.04 \\end{bmatrix}$$

\times \

$$\begin{bmatrix} 150 & 175 & 100 \\\\ 150 & 150 & 125 \\\\ 125 & 250 & 300 \\end{bmatrix}$$

\\)

b. \\( \

$$\begin{bmatrix} 150 & 175 & 100 \\\\ 150 & 150 & 125 \\\\ 125 & 250 & 300 \\end{bmatrix}$$

\times \

$$\begin{bmatrix} 0.03 \\\\ 0.035 \\\\ 0.04 \\end{bmatrix}$$

\\)

c. \\( \

$$\begin{bmatrix} 0.03 & 0.035 & 0.04 \\end{bmatrix}$$

\times \

$$\begin{bmatrix} 150 & 175 & 100 \\\\ 150 & 150 & 125 \\\\ 125 & 250 & 300 \\end{bmatrix}$$

\\)

d. \\( \

$$\begin{bmatrix} 150 & 175 & 100 \\\\ 150 & 150 & 125 \\\\ 125 & 250 & 300 \\end{bmatrix}$$

\times \

$$\begin{bmatrix} 0.03 & 0.035 & 0.04 \\end{bmatrix}$$

\\)

Explanation:

Step1: Recall Matrix Multiplication Rules

For matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. Let's analyze the dimensions:

• The order matrix (let's call it \( A \)) has dimensions \( 3 \times 3 \) (3 rows: Sm, Med, Lg; 3 columns: Jan, Feb, Mar).
• The cost vector (let's call it \( B \)) has costs for Sm, Med, Lg: so it's a \( 1 \times 3 \) matrix (row vector) or \( 3 \times 1 \) (column vector). But we need to multiply to get total cost per month.

To find total cost for each month, we need to multiply the cost of each size (per cup) by the number of each size ordered per month, then sum. So:

• Let the order matrix \( A =

$$\begin{bmatrix} 150 & 175 & 100 \\ 150 & 150 & 125 \\ 125 & 250 & 300 \end{bmatrix}$$

\) (rows: Sm, Med, Lg; columns: Jan, Feb, Mar)

• The cost row vector \( B =

$$\begin{bmatrix} 0.03 & 0.035 & 0.04 \end{bmatrix}$$

\) (columns: Sm, Med, Lg costs)

For matrix multiplication \( B \times A \), the number of columns in \( B \) (3) must equal the number of rows in \( A \) (3). Then the resulting matrix will be \( 1 \times 3 \), where each element is the total cost for Jan, Feb, Mar respectively.

Let's check each option:

• Option A: \( 3 \times 1 \) matrix times \( 3 \times 3 \) matrix. Number of columns in first (1) ≠ rows in second (3). Invalid.
• Option B: \( 3 \times 3 \) matrix times \( 3 \times 1 \) matrix. Result is \( 3 \times 1 \), which would be cost per size per month, not total per month. Not what we want.
• Option C: \( 1 \times 3 \) matrix times \( 3 \times 3 \) matrix. Number of columns in first (3) = rows in second (3). Result is \( 1 \times 3 \), total cost per month. Correct.
• Option D: \( 3 \times 3 \) matrix times \( 1 \times 3 \) matrix. Number of columns in first (3) ≠ rows in second (1). Invalid.

Answer:

C. \(

$$\begin{bmatrix} 0.03 & 0.035 & 0.04 \end{bmatrix}$$

\times

$$\begin{bmatrix} 150 & 175 & 100 \\ 150 & 150 & 125 \\ 125 & 250 & 300 \end{bmatrix}$$

\)