Question

matrix a represents the boxes of apples, oranges, and grapefruit ordered by four different stores each week.
$a = \

$$\begin{bmatrix} 6 & 5 & 2 \\\\ 7 & 7 & 4 \\\\ 8 & 9 & 3 \\\\ 10 & 10 & 5 \\end{bmatrix}$$

$
which matrix represents the total number boxes of each kind of fruit ordered over a twelve week period?
\\(\bigcirc\\) a. $\

$$\begin{bmatrix} 18 & 17 & 14 \\\\ 19 & 19 & 16 \\\\ 20 & 21 & 15 \\\\ 22 & 22 & 17 \\end{bmatrix}$$

$ \LXI0 d. $\

$$\begin{bmatrix} 72 & 5 & 2 \\\\ 84 & 7 & 4 \\\\ 96 & 9 & 3 \\\\ 120 & 10 & 5 \\end{bmatrix}$$

$
\\(\bigcirc\\) b. $\

$$\begin{bmatrix} 72 & 60 & 24 \\\\ 7 & 7 & 4 \\\\ 8 & 9 & 3 \\\\ 10 & 10 & 5 \\end{bmatrix}$$

$ \LXI0 e. $\

$$\begin{bmatrix} 18 & 5 & 2 \\\\ 19 & 7 & 4 \\\\ 20 & 9 & 3 \\\\ 22 & 10 & 5 \\end{bmatrix}$$

$
\\(\bigcirc\\) c. $\

$$\begin{bmatrix} 72 & 60 & 24 \\\\ 84 & 84 & 48 \\\\ 96 & 108 & 36 \\\\ 120 & 120 & 60 \\end{bmatrix}$$

$

Explanation:

Step1: Understand the operation

To find the total number of boxes over 12 weeks, we need to multiply each element of matrix \( A \) by 12 (since each week's order is repeated for 12 weeks). This is scalar multiplication of a matrix, where we multiply each entry \( a_{ij} \) in matrix \( A \) by the scalar 12.

Step2: Multiply each element by 12

• For the first row: \( 6 \times 12 = 72 \), \( 5 \times 12 = 60 \), \( 2 \times 12 = 24 \)
• For the second row: \( 7 \times 12 = 84 \), \( 7 \times 12 = 84 \), \( 4 \times 12 = 48 \)
• For the third row: \( 8 \times 12 = 96 \), \( 9 \times 12 = 108 \), \( 3 \times 12 = 36 \)
• For the fourth row: \( 10 \times 12 = 120 \), \( 10 \times 12 = 120 \), \( 5 \times 12 = 60 \)

So the resulting matrix is:

$$
LATEXBLOCK0
$$

Answer:

C. \(

$$\begin{bmatrix} 72 & 60 & 24 \\ 84 & 84 & 48 \\ 96 & 108 & 36 \\ 120 & 120 & 60 \end{bmatrix}$$

\)