We are given a matrix $C$ representing the sales of small, medium, and large coffees for one week at Faith's coffee shop. We are asked to: a) State the order of matrix $C$. b) Interpret the meaning of the element $C_{11}$. c) Multiply the matrix $[1\ 1\ 1]$ by matrix $C$ and explain the resulting information. d) Given matrix $D$ which represents the sales for the second week, calculate $C+D$.

1. Problem Description

We are given a matrix $C$ representing the sales of small, medium, and large coffees for one week at Faith's coffee shop. We are asked to:

a) State the order of matrix $C$.

b) Interpret the meaning of the element $C_{11}$.

c) Multiply the matrix $[1\ 1\ 1]$ by matrix $C$ and explain the resulting information.

d) Given matrix $D$ which represents the sales for the second week, calculate $C+D$.

2. Solution Steps

a) The matrix $C$ has 3 rows (corresponding to Small, Medium, and Large sizes) and 7 columns (corresponding to Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday). Therefore, the order of matrix $C$ is $3 \times 7$.

b) The element $C_{11}$ refers to the element in the first row and first column of matrix $C$. This represents the number of small coffees sold on Monday. From the matrix $C$, $C_{11} = 24$. So, $C_{11}$ represents the number of small coffees sold on Monday.

c) We need to multiply the matrix $[1\ 1\ 1]$ by matrix $C$. This means we sum the entries in each column of $C$.

$[1\ 1\ 1] \times C = [1\ 1\ 1] \times \begin{bmatrix} 24 & 18 & 17 & 26 & 21 & 25 & 32 \\ 32 & 23 & 23 & 31 & 28 & 34 & 40 \\ 15 & 11 & 14 & 12 & 19 & 26 & 31 \end{bmatrix}$

The resulting matrix will be a $1 \times 7$ matrix:

$[24+32+15\ 18+23+11\ 17+23+14\ 26+31+12\ 21+28+19\ 25+34+26\ 32+40+31] = [71\ 52\ 54\ 69\ 68\ 85\ 103]$

$[71\ 52\ 54\ 69\ 68\ 85\ 103]$ provides the total number of coffees sold each day of the first week (Monday to Sunday).

d) We need to add matrix $C$ and matrix $D$. Matrix $D$ is:

$D = \begin{bmatrix} 22 & 22 & 19 & 22 & 25 & 23 & 34 \\ 29 & 25 & 25 & 32 & 25 & 34 & 37 \\ 12 & 12 & 14 & 16 & 20 & 27 & 32 \end{bmatrix}$

$C + D = \begin{bmatrix} 24 & 18 & 17 & 26 & 21 & 25 & 32 \\ 32 & 23 & 23 & 31 & 28 & 34 & 40 \\ 15 & 11 & 14 & 12 & 19 & 26 & 31 \end{bmatrix} + \begin{bmatrix} 22 & 22 & 19 & 22 & 25 & 23 & 34 \\ 29 & 25 & 25 & 32 & 25 & 34 & 37 \\ 12 & 12 & 14 & 16 & 20 & 27 & 32 \end{bmatrix} = \begin{bmatrix} 46 & 40 & 36 & 48 & 46 & 48 & 66 \\ 61 & 48 & 48 & 63 & 53 & 68 & 77 \\ 27 & 23 & 28 & 28 & 39 & 53 & 63 \end{bmatrix}$

3. Final Answer

a) The order of matrix $C$ is $3 \times 7$.

b) $C_{11}$ represents the number of small coffees sold on Monday.

c) $[1\ 1\ 1] \times C = [71\ 52\ 54\ 69\ 68\ 85\ 103]$. This matrix provides the total number of coffees sold each day of the first week (Monday to Sunday).

d) $C + D = \begin{bmatrix} 46 & 40 & 36 & 48 & 46 & 48 & 66 \\ 61 & 48 & 48 & 63 & 53 & 68 & 77 \\ 27 & 23 & 28 & 28 & 39 & 53 & 63 \end{bmatrix}$