A coffee shop sells coffee in three sizes: small (S), medium (M), and large (L). The sales for one week are shown in matrix $C$. The problem asks to find the order of matrix $C$, the meaning of the element $C_{11}$, multiply the matrix $[1 \ 1 \ 1]$ by the matrix $C$, and explain what the result of the multiplication represents.

1. Problem Description

A coffee shop sells coffee in three sizes: small (S), medium (M), and large (L). The sales for one week are shown in matrix $C$. The problem asks to find the order of matrix $C$, the meaning of the element $C_{11}$, multiply the matrix $[1 \ 1 \ 1]$ by the matrix $C$, and explain what the result of the multiplication represents.

2. Solution Steps

a) The order of matrix $C$ is the number of rows by the number of columns. Matrix $C$ has 3 rows and 7 columns.

b) The element $C_{11}$ represents the element in the first row and the first column of matrix $C$. In this context, it represents the number of small coffees sold on Monday.

c) i) Multiply matrix $[1 \ 1 \ 1]$ by matrix $C$:

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

$= [1(24)+1(32)+1(15) \ 1(18)+1(23)+1(11) \ 1(17)+1(23)+1(14) \ 1(26)+1(31)+1(12) \ 1(21)+1(28)+1(19) \ 1(25)+1(34)+1(26) \ 1(32)+1(40)+1(31)]$

$= [24+32+15 \ 18+23+11 \ 17+23+14 \ 26+31+12 \ 21+28+19 \ 25+34+26 \ 32+40+31]$

$= [71 \ 52 \ 54 \ 69 \ 68 \ 85 \ 103]$

c) ii) The resulting matrix from part i) gives the total number of coffees sold each day of the week. For example, the first element (71) is the total number of coffees (small, medium, and large) sold on Monday.

3. Final Answer

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

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

c) i) $[1 \ 1 \ 1] C = [71 \ 52 \ 54 \ 69 \ 68 \ 85 \ 103]$

c) ii) The solution to part i) provides the total number of coffees (small, medium, and large combined) sold for each day of the week.